# Anomalous convective transport of the tokamak edge plasma, caused by the
inhomogeneous ion cyclotron parametric turbulence

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

In this paper, we develop the kinetic and hydrodynamic theories of the
convective mesoscale flows driven by the spatially inhomogeneous electrostatic
ion cyclotron parametric microturbulence in the pedestal plasma with a sheared
poloidal flow. The developed kinetic theory predicts the generation of the
sheared poloidal convective flow, and of the radial compressed flow with
radial flow velocity gradient. The developed hydrodynamic theory of the
convective flows 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. This gradient density growth is
limited by the formation of the radial oscillating with time ion outflow of
pedestal plasma to scrape-off layer.

###### pacs:

52.35.Ra, 52.35.Kt

##  I Introduction

It was found in the fast wave (FW) heating experimentsPerkins ; Perkins1  on
the National Spherical Torus eXperiment (NSTX) that a significant part of the
FW power loss occurs due to the anomalous convective flow of the collisionless
dense hot plasma from the edge to the cold low density scrape-off layer (SOL)
plasma. In the SOL, this lost hot plasma propagates mostly along SOL magnetic
field lines to the divertor regions. It was estimated in Ref.Mikhailenko ,
where the theory of the ion cyclotron (IC) electrostatic parametric
turbulence, driven by the strong inhomogeneous FW electric field in the
inhomogeneous plasmas, was developed, that the origin of this convective flow
may be the interactions of ions and electrons with radially inhomogeneous
microscale IC and drift turbulence. It was found that the radial and poloidal
velocities of this flow are proportional to the spatial gradient of the
spectral intensity of the turbulent electric field. This convective flow has a
low flow velocity in the plasma core and in the SOL, where the levels of the
IC parametric turbulence and of the drift turbulence are low, and the plasma
parameters are weakly inhomogeneous. The edge transport barrier with steep
density profile (commonly referred to as the pedestal) is, therefore, the most
preferable region for the development of the intensive convective flows. The
transport barrier establishes in the regimes of the enhanced confinement (or
H-mode regime) which is the basic operational regime of tokamaks. This regime
is characterized by the suppression of the drift type instabilities by the
poloidal sheared flow, spontaneously developed in the tokamak edge, with the
flow velocity shearing rate comparable with or above the maximum linear growth
rate of the suppressed instabilities. Note, that the parametric IC
instabilities have the growth rate much above the observed velocity shearing
rate and are not suppressed by this flow. Beginning from the first observation
of the H-mode of operation, it was found that this mode has been accompanied
by periodic bursts of the ejected to SOL plasma structures, which are labelled
as the edge localized modes (ELMs)Zohm ; Leonard  and blobsD'Ippolito  with
considerably larger density than the surrounding low density cold SOL plasma.
It is well recognised now that the non-diffusive transport of the heat and
matter outwards across the magnetic field, which occurs in addition to the
anomalous diffusion, is the universal phenomenonD'Ippolito  experimentally
observed in tokamaks.

It was found in Ref.Mikhailenko , that the radial velocity of the convective
flow is in the direction opposite to the gradient of the spectral intensity of
the microscale turbulence. In the tokamak plasmas, this convective flow moves
outward of pedestal to the SOL region. It may be concluded therefore that the
dynamical processes in the pedestal and near SOL are heavily determined by the
convective flows formed in this region by the inhomogeneous microturbulence
and by interplay of the convective flows with the sheared poloidal edge flow
of a tokamak plasma, which is the inherent component of the pedestal
development and sustainment.

The focus of this paper is the detailed kinetic and hydrodynamic theory of the
non-diffusive convective flows, driven by the spatially inhomogeneous
parametric microturbulence in the sheared poloidal edge (basic) flow of a
tokamak plasma. In Sec. II, we present a detailed description of the two-scale
approach to the kinetic theory of the IC parametric instabilities driven by
the inhomogeneous FW in inhomogeneous plasmas, which is employed in this
paper. In Sec. III, we present the theory of the mesoscale convective flows
driven by the inhomogeneous parametric IC and drift microturbulence in the
poloidal sheared flow. The derived Vlasov equations for ions and electrons,
which govern the mesoscale evolution of the plasma, reveal the existence of
the radial and poloidal convective flows with radially inhomogeneous flow
velocities. Because of the presence of such flows, which make a dominant
contribution to the evolution of inhomogeneous plasma to the equilibrium
state, the conventional theory of the stability of the steady plasma cannot be
applied to this problem. In this paper, the problem of the origin of the
inhomogeneous convective flows and of their temporal evolution is analysed
employing the nonmodal approach, which was developed in our theory of the
temporal evolution and suppression of the instabilities of a plasma sheared
flow across the magnetic fielsMikhailenko1 ; Mikhailenko2 ; Mikhailenko3 .
This approach employs the solution of the initial value problem instead of the
application of the spectral transform in time variable, which is generally
applied in the stability theory for the uniform plasmas with steady uniform
flows. In Sec. III, we develop the nonmodal theory of the convective flows
with the radially inhomogeneous flow velocity, which develops in the poloidal
sheared flow of the plasma edge. The convective flow involves the poloidal
flow with a radially sheared velocity, and the radial compressed/stretched
flow with a radially inhomogeneous flow velocity, the effect of which on the
plasma stability was not considered yet.

In Sec. IV, we derive, as the moments of the electron and ion mesoscale Vlasov
equations, the fluid equations for the plasma with the sheared and
compressed/stretched flows. These equations, which contain the continuity, the
momentum, and the temperature equations for both plasma species, are solved as
the initial value problem which reveal the decisive effect of the compressed
radial flow on the formation of the plasma outflows. Conclusions are presented
in Sec. V.

##  II Two-scale approach to the theory of the parametric instabilities of the
inhomogeneous plasma driven by the inhomogeneous fast wave

Our theory of the microscale turbulence, driven by the inhomogeneous FW in the
inhomogeneous plasmas, and the theory of the mesoscale convective flows,
generated by the inhomogeneous microscale IC and drift turbulence, is based on
the Vlasov-Poisson system of equation. In this theory we use the approximation
of the slab geometry 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 employed in
our theory for the velocity distribution function $F_{\alpha}$ of $\alpha$
plasma species ($\alpha=i$ for ions and $\alpha=e$ for electrons) in
coordinates $\mathbf{r}=\left(x,y,z\right)$,

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

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 respond on FW, the uniform plasma-confining magnetic field
$\mathbf{B}_{0}$ directed along the coordinate $z$, and FW magnetic field
$\mathbf{B}_{1}\left(\mathbf{r},t\right)$. In the edge layer of the tokamak
plasma, the spatial inhomogeneity lengths of $\mathbf{E}_{0x}\left(x\right)$
and of FW fields are commensurable with a spatial inhomogeneity length of the
pedestal plasma density. This mesoscale spatial inhomogeneity length on order
of the pedestal width is much less than the wavelength of FW and the
inhomogeneity lengths of the plasma parameters in the bulk of the tokamak
plasma, but is much larger than the radial wavelength of the IC parametric
microturbulence presented in Eq. (1) by electric field
$\tilde{\mathbf{E}}\left(\mathbf{r},t\right)$. This microturbulence develops
in the inhomogeneous pedestal plasma by the inhomogeneous FW. Therefore,
electric field $\tilde{\mathbf{E}}\left(\mathbf{r},t\right)$ of IC
microrurbulence involves the microscale and mesoscale spatial inhomogenieties.
Thus, the consistent theory which involves the theory of the IC parametric
turbulence of the inhomogeneous plasma, driven by the inhomogeneous FW, and
the theory of the mesoscale plasma evolution caused by the mesoscale
inhomogeneities of the IC microturbulence, should include the treatment of
plasma evolution on the micro- as well as on the mesoscales. We employ for the
development of that theory the well known two - scale method (simple
introduction to the variety versions and applications of this method is
presented in Ref.Nayfeh ), which is used for problems in which the solutions
depend simultaneously on widely different scales. Applying this method to our
theory, we introduce, jointly with variables $\mathbf{r}=\left(x,y,z\right)$
for the microscale fast variations, the spatial $X=\varepsilon x$ variable,
where $\varepsilon\ll 1$ is an artificial dimensionless small parameter, which
distinguishes the slow mesoscale variations along $x$ direction,
$\partial/\partial x=\varepsilon\partial/\partial X$, in the Vlasov equation.

The Vlasov equation (1) in two-scales presentation has a form

$\displaystyle\displaystyle\frac{\partial
F_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)}{\partial
t}+\mathbf{v}\left(\frac{\partial
F_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)}{\partial\mathbf{r}}+\varepsilon\frac{\partial
F_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)}{\partial X}\right)$
$\displaystyle\displaystyle+\frac{e_{\alpha}}{m_{\alpha}}\left(\mathbf{E}_{0x}\left(X\right)+\mathbf{E}_{1}\left(X,t\right)+\tilde{\mathbf{E}}\left(\mathbf{r},X,t\right)\right.$
$\displaystyle\displaystyle\left.+\frac{1}{c}\left[\mathbf{v}\times\left(\mathbf{B}_{0}+\mathbf{B}_{1}\left(X,t\right)\right)\right]\right)\frac{\partial
F_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)}{\partial\mathbf{v}}=0,$ (2)

where FW fields

$\displaystyle\displaystyle\mathbf{E}_{1}\left(X,t\right)=\mathbf{E}_{1x}\left(X\right)\cos\omega_{0}t+\mathbf{E}_{1y}\left(X\right)\sin\omega_{0}t,$
(3)
$\displaystyle\displaystyle\mathbf{B}_{1}\left(X,t\right)=\frac{c}{\omega_{0}}\frac{dE_{1y}\left(X\right)}{dX}\cos\omega_{0}t\,\mathbf{e}_{z}.$
(4)

are inhomogeneous only on the mesoscale. The electric field
$\tilde{\mathbf{E}}\left(\mathbf{r},X,t\right)$, which is inhomogeneous on the
micro- and mesoscale, is governed by the Poisson equation,

$\displaystyle\displaystyle\nabla\cdot\tilde{\mathbf{E}}\left(\mathbf{r},X,t\right)=4\pi\sum_{\alpha=i,e}e_{\alpha}\int
f_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)d{\bf v},$ (5)

in which $f_{\alpha}$ is the fluctuating part of the distribution function
$F_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)$,
$f_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)=F_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)-F_{0\alpha}\left(\mathbf{v},X,t\right)$.
$F_{0\alpha}$ is the equilibrium distribution function of the inhomogeneous
plasma; this function is governed by Eq. (2) with
$\tilde{\mathbf{E}}\left(\mathbf{r},X,t\right)=0$.

The central point in the theory of the IC parametric instabilities driven by
the spatially inhomogeneous FW in the inhomogeneous plasma without sheared
flow, developed Ref. Mikhailenko , is the transformation of the velocity
$\mathbf{v}$, mesoscale coordinate $X$ and microscale coordinates
$\mathbf{r}=\left(x,y,z\right)$ in Eq. (1) to the velocity $\mathbf{v}_{i}$,
the mesoscale coordinate $X_{i}$ and the microscale coordinates
$\mathbf{r}_{i}$, determined in the frame of reverences which moves with
velocity $\mathbf{V}_{i}\left(X_{i},t\right)$ equal to the velocity of an ion
in the plasma-confining magnetic field $\mathbf{B}_{0}$, and FW fields
relative to the laboratory frame. The transformation
$\left(\mathbf{v},X,\mathbf{r},t\right)\rightarrow\left(\mathbf{v}_{i},X_{i},\mathbf{r}_{i},t\right)$
is performed by employing the relation

$\displaystyle\displaystyle\mathbf{v}=\mathbf{v}_{i}+\mathbf{V}_{i}\left(X_{i},t\right),$
(6)

and the inverse relation

$\displaystyle\displaystyle\mathbf{v}_{i}=\mathbf{v}-\mathbf{V}\left(X,t\right);$
(7)

the relation

$\displaystyle\displaystyle
X=X_{i}+R_{ix}\left(X_{i},t\right)=X_{i}+\int\limits^{t}_{t_{0}}V_{ix}\left(X_{i},t_{1}\right)dt_{1},$
(8)

and the inverse relation,

$\displaystyle\displaystyle
X_{i}=X-\int\limits^{t}_{t_{0}}V_{x}\left(X,t_{1}\right)dt_{1}.$ (9)

The velocity $\mathbf{V}_{i}\left(X_{i},t\right)$ is determined by the
equationMikhailenko

$\displaystyle\displaystyle\frac{d\mathbf{V}_{i}}{dt}=\frac{e_{i}}{m_{\alpha}}\left(\mathbf{E}_{1}\left(X_{i}+R_{ix}\left(X_{i},t\right),t\right)+\frac{1}{c}\left[\mathbf{V}_{i}\times\mathbf{B}_{0}\right]\right.$
$\displaystyle\displaystyle\left.+\frac{1}{c}\left[\mathbf{V}_{i}\times\mathbf{B}_{1}\left(X_{i}+R_{ix}\left(X_{i},t\right),t\right)\right]\right),$
(10)

with initial value $\mathbf{V}_{i}\left(X_{i},t=t_{0}\right)=0$. In this
equation, the variable $X_{i}$, which determines the spatial inhomogeneity of
the plasma and FW field, enters to Eq. (10) as a parameter. It was derived in
Ref.Mikhailenko  that when the amplitude of the ion displacement
$\left|R_{ix}\left(X_{i},t\right)\right|$ in FW field is much less than the
spatial scale length $L_{E}$ of the radial inhomogeneity of the FW field, the
solution to Eq. (10) may be derived in a form of a power series expansion in
powers of $R_{ix}/L_{E}$. It was found Mikhailenko  that when
$\omega_{0}-\omega_{ci}\sim\omega_{ci}$, where $\omega_{ci}$ is the ion
cyclotron frequency, the terms on the order of $O\left(R_{ix}/L_{E}\right)$ of
this expansions with FW for pedestal parameters are negligibly small and may
be neglected. It was derived in Ref.Mikhailenko  that with variables
$\mathbf{v}_{i}$, $X_{i}$, and $\mathbf{r}_{i}$, where

$\displaystyle\displaystyle\mathbf{r}=\mathbf{r}_{i}+\mathbf{R}_{i}\left(X_{i},t\right)=\mathbf{r}_{i}+\int\limits^{t}_{t_{0}}\mathbf{V}_{i}\left(X_{i},t\right)dt_{1},$
(11)

FW electric field is presented in the Vlasov equation (2) for the microscale
perturbations only in terms of the order of $|R_{ix}/L_{E}|\ll 1$ which may be
neglected. Therefore the Vlasov equation for
$F_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)$ with great accuracy
has a form as for a steady plasma in the uniform magnetic field
$\mathbf{B}_{0}$ without FW fields, and contains the variable $X_{i}$ as a
parameter, i. e.

$\displaystyle\displaystyle\frac{\partial
F_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)}{\partial
t}+\mathbf{v}_{i}\frac{\partial
F_{i}}{\partial\mathbf{r}_{i}}+\frac{e_{i}}{m_{i}c}\left[\mathbf{v}_{i}\times\mathbf{B}_{0}\right]\frac{\partial
F_{i}}{\partial\mathbf{v}_{i}}$
$\displaystyle\displaystyle+\frac{e_{i}}{m_{i}}\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\frac{\partial
F_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)}{\partial\mathbf{v}_{i}}=0.$
(12)

In fact, this equation is the result of the quantitative
justificationMikhailenko  of the application of the local approximation to the
theory of the parametric instabilities driven by the inhomogeneous FW field in
the pedestal plasma. This result is completely applicable to the case of FW in
the poloidal sheared flow of a tokamak plasma. In this case, velocity
$\mathbf{V}_{i}$ is determined by Eq. (10) with the electric field
$\mathbf{E}_{1}\left(X_{i}+R_{ix}\left(X_{i},t\right),t\right)+\mathbf{E}_{0x}\left(X_{i}+R_{ix}\left(X_{i},t\right)\right)$.

The equation similar to Eq. (12) is derived for the Vlasov equation for the
electron distribution function
$F_{e}\left(\mathbf{v}_{e},\mathbf{r}_{e},X_{e},t\right)$, where
$\mathbf{v}_{e}$ and $\mathbf{r}_{e}$ are the velocity and the position vector
of electron in the frame of references which moves with velocity
$\mathbf{V}_{e}\left(X_{e},t\right)$ determined by Eq. (10) with ion species
subscript $i$ changed on the electron subscript $e$. It follows from Eq. (12)
that the Vlasov equation for the perturbation
$f_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)$ of the ion
distribution function $F_{i0}\left(\mathbf{v}_{i},X_{i}\right)$ and the
similar equation for the perturbation
$f_{e}\left(\mathbf{v}_{e},\mathbf{r}_{e},X_{e},t\right)$ of the electron
distribution function $F_{e0}\left(\mathbf{v}_{e},X_{e}\right)$ are the same
as for a plasma without FW field,

$\displaystyle\displaystyle\frac{\partial
f_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)}{\partial
t}+\mathbf{v}_{i}\frac{\partial
f_{i}}{\partial\mathbf{r}_{i}}+\frac{e_{i}}{m_{i}c}\left[\mathbf{v}_{i}\times\mathbf{B}_{0}\right]\frac{\partial
f_{i}}{\partial\mathbf{v}_{i}}$
$\displaystyle\displaystyle+\frac{e_{i}}{m_{i}}\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\frac{\partial
f_{i}}{\partial\mathbf{v}_{i}}=-\frac{e_{i}}{m_{i}}\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\frac{\partial
F_{i0}\left(\mathbf{v}_{i},X_{i}\right)}{\partial\mathbf{v}_{i}},$ (13)
$\displaystyle\displaystyle\frac{\partial
f_{e}\left(\mathbf{v}_{e},\mathbf{r}_{e},X_{e},t\right)}{\partial
t}+\mathbf{v}_{e}\frac{\partial
f_{e}}{\partial\mathbf{r}_{e}}+\frac{e}{m_{e}c}\left[\mathbf{v}_{e}\times\mathbf{B}_{0}\right]\frac{\partial
f_{e}}{\partial\mathbf{v}_{e}}$
$\displaystyle\displaystyle+\frac{e}{m_{e}}\tilde{\mathbf{E}}_{e}\left(\mathbf{r}_{e},X_{e},t\right)\frac{\partial
f_{e}}{\partial\mathbf{v}_{e}}=-\frac{e}{m_{e}}\tilde{\mathbf{E}}_{e}\left(\mathbf{r}_{e},X_{e},t\right)\frac{\partial
F_{e0}\left(\mathbf{v}_{e},X_{e}\right)}{\partial\mathbf{v}_{e}}.$ (14)

The mesoscale spatial variables $X_{i}$ and $X_{e}$ are introduced to Eqs.
(13), (14) by the spatial inhomogeneity of
$F_{i0}\left(\mathbf{v}_{i},X_{i}\right)$ and
$F_{e0}\left(\mathbf{v}_{e},X_{e}\right)$ functions and present in these
equations as a parameters. Thus, the solutions to $f_{i}$ and $f_{e}$ also
depend on $X_{i}$ and $X_{e}$. The solutions to Eqs. (13) and (14) were
derived in Ref.Mikhailenko , where the local approximation to the theory of
the parametric IC microinstabilities driven by the inhomogeneous FW in the
inhomogeneous plasma was developed. These results for $f_{i}$ and $f_{e}$, as
well as ones for the perturbed ion and electron densities, are applicable to
the present investigations of the IC microinstabilities in the poloidal
sheared flow.

The Vlasov equations (13) and (14) for
$f_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)$ and
$f_{e}\left(\mathbf{v}_{e},\mathbf{r}_{e},X_{e},t\right)$, and the Poisson
equation

$\displaystyle\displaystyle\nabla\cdot\tilde{\mathbf{E}}\left(\mathbf{r}_{i},X_{i},t\right)=4\pi\sum_{\alpha=i,e}e_{\alpha}\int
f_{\alpha}\left(\mathbf{v}_{\alpha},\mathbf{r}_{\alpha},X_{\alpha},t\right)d\mathbf{v}_{\alpha},$
(15)

for the electric field
$\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)=-\frac{\partial\varphi_{i}\left(\mathbf{r}_{i},X_{i},t\right)}{\partial\mathbf{r}_{i}}$
compose the system of equation for the investigation of the microscale IC
parametric instabilities of a plasma sheared flow in the inhomogeneous FW
field, and the nonlinear evolution and saturation these instabilities. In the
saturation state, the electric field $\tilde{\mathbf{E}}$ of the electrostatic
parametric microturbulence is determined in the ion frame in the form

$\displaystyle\displaystyle\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)=\int
d\mathbf{k}\tilde{\mathbf{E}}\left(\mathbf{k},X_{i}\right)e^{-i\omega\left(\mathbf{k},X_{i}\right)t+i\mathbf{k}\mathbf{r}_{i}+i\theta\left(\mathbf{k}\right)}$
$\displaystyle\displaystyle=-i\int
d\mathbf{k}\mathbf{k}\varphi\left(\mathbf{k},X_{i}\right)e^{-i\omega\left(\mathbf{k},X_{i}\right)t+i\mathbf{k}\mathbf{r}_{i}+i\theta\left(\mathbf{k}\right)},$
(16)

where the integration over $\mathbf{k}$ is performed over wave numbers 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. (16) 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

$\displaystyle\displaystyle\tilde{\mathbf{E}}_{e}\left(\mathbf{r}_{e},X_{e},t\right)=\int
d\mathbf{k}\tilde{\mathbf{E}}_{e}\left(\mathbf{k},X_{e}\right)e^{-i\omega\left(\mathbf{k},X_{e}\right)t+i\mathbf{k}\mathbf{r}_{e}+i\theta\left(\mathbf{k}\right)}$
$\displaystyle\displaystyle=\int
d\mathbf{k}\tilde{\mathbf{E}_{i}}\left(\mathbf{k},X_{i}\right)e^{-i\omega\left(\mathbf{k},X_{i}\right)t+i\mathbf{k}\mathbf{r}_{i}+i\mathbf{k}\left(\mathbf{r}_{e}-\mathbf{r}_{i}\right)+i\theta\left(\mathbf{k}\right)}$
$\displaystyle\displaystyle=\int
d\mathbf{k}\tilde{\mathbf{E}}_{i}\left(\mathbf{k},X_{i}\right)\sum\limits_{p=-\infty}^{\infty}J_{p}\left(a_{ie}\right)e^{i\mathbf{k}\mathbf{r}_{i}-i\Omega_{p}\left(\mathbf{k},X_{i}\right)t-ip\delta_{ie}\left(\mathbf{k},X_{i}\right)+i\theta\left(\mathbf{k}\right)},$
(17)

where

$\displaystyle\displaystyle\Omega_{p}\left(\mathbf{k},X_{i}\right)=\omega\left(\mathbf{k},X_{i}\right)+p\omega_{0},$
(18)

and $J_{p}\left(a_{ie}\right)$ 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 .

The theory of the parametric microturbulence, developed in details in
Ref.Mikhailenko  for the case $V^{\prime}_{0}=0$, is completely applicable to
the tokamak poloidal sheared flows, where always the velocity shearing rate
$V^{\prime}_{0}=-cE^{\prime}_{0}/B_{0}$ is much less than IC frequency
$\omega_{ci}$ and is less than the growth rate of the IC parametric
instabilities. However, the poloidal sheared flow with the velocity shearing
rate commensurable with velocity gradients of the inhomogeneous mesoscale
convective flows strongly affects their structure and temporal evolution.

##  III Mesoscale convective flows in the poloidally sheared plasma flow

On the nonlinear stage of the IC parametric microturbulence evolution at 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 (16) 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. Here we consider
the average effect of the mesoscale inhomogeneity of the microscale IC
turbulence on the mesoscale evolution of the ion and electron distribution
functions. For this goal, we transform velocity $\mathbf{v}_{i}$ and mesoscale
$X_{i}$, $Y_{i}$ and microscale $\mathbf{r}_{i}$ spatial variables to new
velocity $\tilde{\mathbf{v}}_{i}$ and coordinates $\tilde{X}_{i}$,
$\tilde{Y}_{i}$, $\tilde{\mathbf{r}}_{i}$, in which the thermal ion motion and
the ion motion in the turbulent electric field $\tilde{\mathbf{E}}_{i}$ are
separated. The transformation
$\left(\mathbf{v}_{i},X_{i},Y_{i},\mathbf{r}_{i},t\right)\rightarrow\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},\tilde{\mathbf{r}}_{i},t\right)$
is determined by the relations

$\displaystyle\displaystyle\tilde{\mathbf{v}}_{i}=\mathbf{v}_{i}-\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right),$
(19)
$\displaystyle\displaystyle\tilde{X}_{i}=X_{i}-\int\limits^{t}_{t_{0}}\tilde{V}_{ix}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1},$
(20)
$\displaystyle\displaystyle\tilde{Y}_{i}=Y_{i}-\int\limits^{t}_{t_{0}}\tilde{V}_{iy}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1},$
(21)

and

$\displaystyle\displaystyle\tilde{\mathbf{r}}_{i}=\mathbf{r}_{i}-\int\limits^{t}_{t_{0}}\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1},$
(22)

or by their inverse,

$\displaystyle\displaystyle\mathbf{v}_{i}=\tilde{\mathbf{v}}_{i}+\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),$
(23) $\displaystyle\displaystyle
X_{i}=\tilde{X}_{i}+\int\limits^{t}_{t_{0}}\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1},$
(24) $\displaystyle\displaystyle
Y_{i}=Y-V^{\prime}_{0}Xt=\tilde{Y}_{i}+\int\limits^{t}_{t_{0}}\tilde{U}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1},$
(25)

and

$\displaystyle\displaystyle\mathbf{r}_{i}=\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)=\tilde{\mathbf{r}}_{i}+\int\limits^{t}_{t_{0}}\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}.$
(26)

The velocity $\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)$ is
determined by the Euler equation

$\displaystyle\displaystyle\frac{\partial\tilde{\mathbf{V}}_{i}}{\partial
t}+\tilde{V}_{ix}\frac{\partial\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)}{\partial{X}_{i}}=\frac{e_{i}}{m_{i}}\left(\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)+\frac{1}{c}\left[\tilde{\mathbf{V}}_{i}\times\mathbf{B}_{0}\right]\right)$
(27)

as the velocity of an ion in the electric field
$\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)$ of the IC
parametric turbulence, where variables $\mathbf{r}_{i},X_{i}$ are determined
in the frame of references which moves with the velocity of an ion in the
field of the fast wave in a poloidal sheared flow. In variables
$\left(\tilde{X}_{i},t\right)$, 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. (27) vanishes and this operator is transformed to the linear one,
$\frac{\partial}{\partial t}$. Then, Eq. (27) becomes the ordinary
differential equation

$\displaystyle\displaystyle\frac{d}{dt}\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)$
$\displaystyle\displaystyle=\frac{e_{i}}{m_{i}}\left(\tilde{\mathbf{E}}_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),\tilde{X}_{i}+\tilde{R}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),t\right)+\frac{1}{c}\left[\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\times\mathbf{B}_{0}\right]\right).$
(28)

for
$\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)=\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)$,
where $\mathbf{r}_{i},X_{i}$ are determined as a functions of
$\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t$ by Eqs. (26) and (24). The spatial
micro- and mesoscale variables $\tilde{\mathbf{r}}_{i}$ and $\tilde{X}_{i}$
enter to Eq. (28) as parameters. Equation (28), as well as Eq. (10), reveals
as a hybrid of Eulerian - Lagrangian description of the ion dynamics, in which
the ion velocity and the ion displacement associated with the waves are
defined as in the Eulerian dynamics by the functions of $\tilde{X}_{i}$ and
$t$ and not primarily as a function of the initial position of ion as in a
purely Lagrangian description. At the same time, the nonlinear convective
derivative is absent in Eqs. (28) and (10).

Using the expansion for the electric field $\tilde{\mathbf{E}}_{i}$ in Eq.
(28),

$\displaystyle\displaystyle\tilde{\mathbf{E}}_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),\tilde{X}_{i}+\tilde{R}_{ix}\left(\mathbf{\tilde{r}}_{i},\tilde{X}_{i},t\right),t\right)\approx\tilde{\mathbf{E}}_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),\tilde{X}_{i},t\right)$
$\displaystyle\displaystyle+\tilde{R}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\frac{\partial}{\partial\tilde{X}_{i}}\tilde{\mathbf{E}}_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),\tilde{X}_{i},t\right),$
(29)

which is valid for the small displacement,
$\left|\tilde{R}_{ix}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t\right)\right|\ll
L_{\tilde{E}}$, of an ion in the inhomogeneous electric field
$\tilde{\mathbf{E}}_{i}$, the solution to Eq. (28) with the initial value
$\tilde{\mathbf{U}}_{i}\left(\mathbf{\tilde{r}}_{i},\tilde{X}_{i},t_{0}\right)=0$
is easily derived and may be presented in the form

$\displaystyle\displaystyle\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\approx\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)+\tilde{U}^{(1)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),$
(30)

and

$\displaystyle\displaystyle\tilde{U}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\approx\tilde{U}^{(0)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)+\tilde{U}^{(1)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right).$
(31)

In Eqs. (30), (31)

$\displaystyle\displaystyle\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)=\frac{e_{i}}{m_{i}}\int\limits^{t}_{t_{0}}dt_{1}\left[\tilde{E}_{x}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\cos\omega_{ci}\left(t-t_{1}\right)\right.$
$\displaystyle\displaystyle\left.+\tilde{E}_{y}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\sin\omega_{ci}\left(t-t_{1}\right)\right],$
(32)
$\displaystyle\displaystyle\tilde{U}^{(0)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)=\frac{e_{i}}{m_{i}}\int\limits^{t}_{t_{0}}dt_{1}\left[-\tilde{E}_{x}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\sin\omega_{ci}\left(t-t_{1}\right)\right.$
$\displaystyle\displaystyle\left.+\tilde{E}_{y}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\cos\omega_{ci}\left(t-t_{1}\right)\right],$
(33)
$\displaystyle\displaystyle\tilde{U}^{(1)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)$
$\displaystyle\displaystyle=\frac{e_{i}}{m_{i}}\int\limits^{t}_{t_{0}}dt_{1}\left[\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}_{x}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\right)\cos\omega_{ci}\left(t-t_{1}\right)\int\limits^{t_{1}}_{t_{0}}dt_{2}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{2}\right)\right.$
$\displaystyle\displaystyle\left.+\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}_{y}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\right)\sin\omega_{ci}\left(t-t_{1}\right)\int\limits^{t_{1}}_{t_{0}}dt_{2}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{2}\right)\right],$
(34)
$\displaystyle\displaystyle\tilde{U}^{(1)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)$
$\displaystyle\displaystyle=\frac{e_{i}}{m_{i}}\int\limits^{t}_{t_{0}}dt_{1}\left[-\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}_{x}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\right)\sin\omega_{ci}\left(t-t_{1}\right)\int\limits^{t_{1}}_{t_{0}}dt_{2}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{2}\right)\right.$
$\displaystyle\displaystyle\left.+\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}_{y}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{1}\right)\right)\cos\omega_{ci}\left(t-t_{1}\right)\int\limits^{t_{1}}_{t_{0}}dt_{2}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t_{2}\right)\right].$
(35)

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

$\displaystyle\displaystyle\frac{\partial F_{i}}{\partial
t}+\tilde{v}_{ix}\frac{\partial
F_{i}}{\partial\tilde{X}_{i}}+\left(\tilde{v}_{iy}-V^{\prime}_{0}t\tilde{v}_{ix}\right)\frac{\partial
F_{i}}{\partial\tilde{Y}_{i}}-\tilde{v}_{ix}\int\limits_{t_{0}}^{t}\frac{\partial}{\partial
X_{i}}\tilde{V}_{ix}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1}\frac{\partial
F_{i}}{\partial\tilde{X}_{i}}$
$\displaystyle\displaystyle-\tilde{v}_{ix}\int\limits_{t_{0}}^{t}\frac{\partial}{\partial
X_{i}}\tilde{V}_{iy}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1}\frac{\partial
F_{i}}{\partial\tilde{Y}_{i}}-\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial
X_{i}}\tilde{V}_{ix}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1}\frac{\partial
F_{i}}{\partial\tilde{X}_{i}}$
$\displaystyle\displaystyle-\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\left(V^{\prime}_{0}t+\int\limits_{t_{0}}^{t}\frac{\partial}{\partial
X_{i}}\tilde{V}_{iy}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1}\right)\frac{\partial
F_{i}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle+\omega_{ci}\tilde{v}_{iy}\frac{\partial
F_{i}}{\partial\tilde{v}_{ix}}-\left(\omega_{ci}+V^{\prime}_{0}\right)\tilde{v}_{ix}\frac{\partial
F_{i}}{\partial\tilde{v}_{iy}}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\left(\frac{\partial}{\partial\tilde{X}_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)-V^{\prime}_{0}t\frac{\partial}{\partial\tilde{Y}_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)\right)\frac{\partial
F_{i}}{\partial\tilde{v}_{ix}}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\frac{\partial}{\partial\tilde{Y}_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)\frac{\partial
F_{i}}{\partial\tilde{v}_{iy}}-\frac{e_{i}}{m_{i}}\frac{\partial}{\partial\tilde{z}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)\frac{\partial
F_{i}}{\partial v_{iz}}=0.$ (36)

The electrostatic potential $\varphi_{i}$ in Eq. (36) depends on the micro-
and mesoscales and can be expressed in the form

$\displaystyle\displaystyle\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)=\tilde{\varphi}_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t\right)+\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right),$
(37)

where $\tilde{\varphi}_{i}$ is the potential of the microscale turbulence, and
$\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$ is the potential of the
electrostatic mesoscale perturbations. The potential
$\varphi\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)$,
averaged over the initial phases of the microscale perturbations, is

$\displaystyle\displaystyle\langle\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)\rangle=\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right),$
(38)

because
$\langle\tilde{\varphi}_{i}\left(\tilde{\mathbf{r}},\tilde{X}_{i},t\right)\rangle=0$.
It follows from Eq. (27), that the equation for the Eulerian mean velocity
$\left\langle\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\right\rangle$
averaged over the initial phases $\theta\left(\mathbf{k}\right)$ ,

$\displaystyle\displaystyle\frac{\partial\left\langle\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\right\rangle}{\partial
t}-\frac{e_{i}}{m_{i}c}\left[\left\langle\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\right\rangle\times\mathbf{B}_{0}\right]$
$\displaystyle\displaystyle=-\left\langle\tilde{V}_{ix}\frac{\partial\tilde{\mathbf{V}}_{i}}{\partial
X_{i}}\right\rangle+\frac{e_{i}}{m_{i}}\left\langle\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\right\rangle,$
(39)

has nonlinear ”extra term” originated from the convective part of the material
time derivative. This term is absent in the averaged hybrid Eulerian -
Lagrangian Eq. (28) for
$\mathbf{U}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)$,

$\displaystyle\displaystyle\frac{d}{dt}\left\langle\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\right\rangle-\frac{e_{i}}{m_{i}c}\left[\left\langle\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\right\rangle\times\mathbf{B}_{0}\right]$
$\displaystyle\displaystyle=\frac{e_{i}}{m_{i}}\left\langle\tilde{\mathbf{E}}_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i}+\tilde{R}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right),t\right)\right\rangle,$
(40)

Contrary to the Eulerian equation, where the average is determined at definite
$X_{i}$ value, in the hybrid of Eulerian - Lagrangian description the average
at time $t$ is taken with respect to the positions
$\tilde{X}_{i}+\tilde{R}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)$
displaced by the microturbulence. The Vlasov equation for the ion distribution
function
$\bar{F}_{i}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)$,
averaged over the microscale initial phases for a time
$t\gg\tau_{corr}\sim\gamma^{-1}$, is

$\displaystyle\displaystyle\frac{\partial\bar{F}_{i}}{\partial
t}+\tilde{v}_{ix}\frac{\partial\bar{F}_{i}}{\partial\tilde{X}_{i}}+\left(\tilde{v}_{iy}-V^{\prime}_{0}t\tilde{v}_{ix}\right)\frac{\partial\bar{F}_{i}}{\partial\tilde{Y}_{i}}-\bar{U}_{ix}\left(\tilde{X}_{i},t\right)\frac{\partial\bar{F}_{i}}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i},t\right)\frac{\partial\bar{F}_{i}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle+\omega_{ci}\tilde{v}_{iy}\frac{\partial\bar{F}_{i}}{\partial\tilde{v}_{ix}}-\left(\omega_{ci}+V^{\prime}_{0}\right)\frac{\partial\bar{F}_{i}}{\partial\tilde{v}_{iy}}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\left(\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial\tilde{X}_{i}}-V^{\prime}_{0}t\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial\tilde{Y}_{i}}\right)\frac{\partial
F_{i}}{\partial\tilde{v}_{ix}}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial\tilde{Y}_{i}}\frac{\partial
F_{i}}{\partial\tilde{v}_{iy}}-\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial\tilde{z}}\frac{\partial
F_{i}}{\partial v_{iz}}=0.$ (41)

In this equation, we use the relations

$\displaystyle\displaystyle\frac{\partial}{\partial
X_{i}}\tilde{V}_{ix}\left(\mathbf{r}_{i},X_{i},t_{1}\right)=\frac{\partial}{\partial
X_{i}}\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)$
$\displaystyle\displaystyle\approx\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)\right)\left(1-\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right)$
(42)

and

$\displaystyle\displaystyle\frac{\partial}{\partial
X_{i}}\tilde{V}_{iy}\left(\mathbf{r}_{i},X_{i},t_{1}\right)=\frac{\partial}{\partial
X_{i}}\tilde{U}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)$
$\displaystyle\displaystyle\approx\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{U}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)\right)\left(1-\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right).$
(43)

These relations stem from the identity
$\tilde{\mathbf{V}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)=\tilde{\mathbf{U}}_{i}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)$,
which follows from Eqs. (19) and (23), and from the identity

$\displaystyle\displaystyle\frac{\partial\tilde{X}_{i}}{\partial
X_{i}}=1-\int\limits^{t}_{t_{0}}dt_{1}\frac{\partial\tilde{V}_{ix}\left(\mathbf{r}_{i},X_{i},t_{1}\right)}{\partial
X_{i}},$

which follows from Eqs. (20) and (24). In Eq. (41),
$\bar{U}_{ix}\left(\tilde{X}_{i},t\right)$ and
$\bar{U}_{iy}\left(\tilde{X}_{i},t\right)$ are the spatially inhomogeneous
velocities of the convective flows along directions of $x_{i}$ and $y_{i}$,
respectively. The velocity $\bar{U}_{ix}\left(\tilde{X}_{i},t\right)$ with
accounting for the terms on the order of
$O\left(\left|\tilde{R}_{ix}/L_{\tilde{E}}\right|^{3}\right)$ is determined by
the equation

$\displaystyle\displaystyle\bar{U}_{ix}\left(\tilde{X}_{i},t\right)=\left\langle\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial
X_{i}}\tilde{V}_{ix}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1}\right\rangle$
$\displaystyle\displaystyle\approx\bar{U}^{(0)}_{ix}\left(\tilde{X}_{i},t\right)+\bar{U}^{(1)}_{ix}\left(\tilde{X}_{i},t\right)+\bar{U}^{(2)}_{ix}\left(\tilde{X}_{i},t\right),$
(44)

where

$\displaystyle\displaystyle\bar{U}^{(0)}_{ix}\left(\tilde{X}_{i},t\right)=\left\langle\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right\rangle,$
(45)
$\displaystyle\displaystyle\bar{U}^{(1)}_{ix}\left(\tilde{X}_{i},t\right)=\left\langle\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(1)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right\rangle$
$\displaystyle\displaystyle+\left\langle\tilde{U}^{(1)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right\rangle,$
(46)
$\displaystyle\displaystyle\bar{U}^{(2)}_{ix}\left(\tilde{X}_{i},t\right)=-\left\langle\tilde{U}^{(0)}_{ix}\left(\mathbf{\tilde{r}}_{i},\tilde{X}_{i},t\right)\right.$
$\displaystyle\displaystyle\left.\times\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\int\limits_{t_{0}}^{t_{1}}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{2}\right)dt_{2}\right\rangle.$
(47)

Velocity $\bar{U}_{iy}\left(\tilde{X}_{i},t\right)$ is expressed as

$\displaystyle\displaystyle\bar{U}_{iy}\left(\tilde{X}_{i},t\right)=\left\langle\tilde{U}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial
X_{i}}\tilde{V}_{iy}\left(\mathbf{r}_{i},X_{i},t_{1}\right)dt_{1}\right\rangle$
$\displaystyle\displaystyle\approx\bar{U}^{(0)}_{iy}\left(\tilde{X}_{i},t\right)+\bar{U}^{(1)}_{iy}\left(\tilde{X}_{i},t\right)+\bar{U}^{(2)}_{iy}\left(\tilde{X}_{i},t\right),$
(48)

where

$\displaystyle\displaystyle\bar{U}^{(0)}_{iy}\left(\tilde{X}_{i},t\right)=\left\langle\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right\rangle,$
(49)
$\displaystyle\displaystyle\bar{U}^{(1)}_{iy}\left(\tilde{X}_{i},t\right)=\left\langle\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(1)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right\rangle$
$\displaystyle\displaystyle+\left\langle\tilde{U}^{(1)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\right\rangle,$
(50)
$\displaystyle\displaystyle\bar{U}^{(2)}_{iy}\left(\tilde{X}_{i},t\right)=-\left\langle\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t\right)\right.$
$\displaystyle\displaystyle\left.\times\int\limits_{t_{0}}^{t}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{iy}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{1}\right)dt_{1}\int\limits_{t_{0}}^{t_{1}}\frac{\partial}{\partial\tilde{X}_{i}}\tilde{U}^{(0)}_{ix}\left(\tilde{\mathbf{r}}_{i},\tilde{X}_{i},t_{2}\right)dt_{2}\right\rangle.$
(51)

For the electric field $\tilde{\mathbf{E}}_{i}$, given by Eq. (16), the
velocities $\bar{U}^{(0)}_{ix}\left(\tilde{X}_{i},t\right)$ and
$\bar{U}^{(0)}_{iy}\left(\tilde{X}_{i},t\right)$ are determined in the
Appendix.

The Vlasov equation for the average electron distribution
$\bar{F}_{e}\left(\tilde{\mathbf{v}}_{e},\tilde{X}_{e},\tilde{Y}_{e},t\right)$,
where $\tilde{X}_{e},\tilde{Y}_{e}$ are determined by Eqs. (19) - (25) with
ion species subscript changed on electron, has a form similar to Eq. (41). The
velocities $\bar{U}_{ex}\left(\tilde{X}_{e},t\right)$ and
$\bar{U}_{ey}\left(\tilde{X}_{e},t\right)$ are determined in this equation by
Eqs. (44) - (53) 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. (30) - (35) with the turbulent electric field
$\tilde{\mathbf{E}}_{e}\left(\tilde{\mathbf{r}}_{e},\tilde{X}_{e},t\right)$,
determined by Eq. (17) .

The Vlasov equations (41) for
$\bar{F}_{i}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)$,
and the similar equation for
$\bar{F}_{e}\left(\tilde{\mathbf{v}}_{e},\tilde{X}_{e},\tilde{Y}_{e},t\right)$,
and the Poisson equation for the potential
$\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$,

$\displaystyle\displaystyle\frac{\partial^{2}\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial^{2}\tilde{X_{i}}}+\frac{\partial^{2}\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial^{2}\tilde{Y_{i}}}=-4\pi\left(e_{i}\int
d\mathbf{v}_{i}\bar{F}_{i}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)\right.$
$\displaystyle\displaystyle\left.-|e|\int
d\mathbf{v}_{e}\bar{F}_{e}\left(\tilde{\mathbf{v}}_{e},\tilde{X}_{e},\tilde{Y}_{e},t\right)\right),$
(52)

compose the Vlasov-Poisson system, which governs the kinetic mesoscale
evolution of a plasma under the average action of the spatially inhomogeneous
microturbulence.

##  IV Hydrodynamics of the mesoscale convective flows

The mesoscale spatial variations of a plasma are involved in the theory of the
microturbulence through the dependences on $X_{i}$ and on $X_{e}$ of the ion
and electron densities and temperatures. In this Section, we derive a closed
set of equations which determine the mesoscale evolution of the ion and
electron densities, velocities, temperatures and of the electrostatic
potential in the poloidal sheared flow with radially inhomogeneous convective
flows. These equations are the first three moments of the Vlasov equations
(41) for $\bar{F}_{i}$ and $\bar{F}_{e}$. Starting with Eq. (41) we derive the
moment equations as follows. By integrating over velocities
$\tilde{\mathbf{v}}_{i}$ we derive the ion particles density conservation
equation:

$\displaystyle\displaystyle\frac{\partial n_{i}}{\partial
t}+\frac{\partial}{\partial\tilde{X}_{i}}\left(n_{i}u_{ix}\right)+\frac{\partial}{\partial\tilde{Y}_{i}}\left(n_{i}\left(u_{iy}-V^{\prime}_{0}tu_{ix}\right)\right)$
$\displaystyle\displaystyle-\bar{U}_{ix}\left(\tilde{X}_{i},t\right)\frac{\partial
n_{i}}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i},t\right)\frac{\partial
n_{i}}{\partial\tilde{Y}_{i}}=0.$ (53)

By first multiplying by $\tilde{\mathbf{v}}_{i}$ of Eq. (41) and integrating
over velocities $\tilde{\mathbf{v}}_{i}$, we obtain the momentum conservation
equations:

$\displaystyle\displaystyle\frac{\partial u_{ix}}{\partial
t}+u_{ix}\frac{\partial
u_{ix}}{\partial\tilde{X}_{i}}+\left(u_{iy}-V^{\prime}_{0}tu_{ix}\right)\frac{\partial
u_{ix}}{\partial\tilde{Y}_{i}}-\bar{U}_{ix}\left(\tilde{X}_{i},t\right)\frac{\partial
u_{ix}}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i},t\right)\frac{\partial
u_{ix}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle=-\frac{1}{m_{i}n_{i}}\left(\frac{\partial
P_{i}}{\partial\tilde{X}_{i}}-V^{\prime}_{0}t\frac{\partial
P_{i}}{\partial\tilde{Y}_{i}}\right)-\frac{e_{i}}{m_{i}}\left(\frac{\partial\Phi\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial\tilde{X}_{i}}-V^{\prime}_{0}t\frac{\partial\Phi\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial\tilde{Y}_{i}}\right)+\omega_{ci}u_{iy},$
(54) $\displaystyle\displaystyle\frac{\partial u_{iy}}{\partial
t}+u_{ix}\frac{\partial
u_{iy}}{\partial\tilde{X}_{i}}+\left(u_{iy}-V^{\prime}_{0}tu_{ix}\right)\frac{\partial
u_{iy}}{\partial\tilde{Y}_{i}}-\bar{U}_{ix}\left(\tilde{X}_{i},t\right)\frac{\partial
u_{iy}}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i},t\right)\frac{\partial
u_{iy}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle=-\frac{1}{m_{i}n_{i}}\frac{\partial
P_{i}}{\partial\tilde{Y}_{i}}-\frac{e_{i}}{m_{i}}\frac{\partial\Phi\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}{\partial\tilde{Y}_{i}}-\left(\omega_{ci}+V^{\prime}_{0}\right)u_{ix}.$
(55)

By first multiplying Eq. (41) by
$\frac{1}{2}\left|\tilde{\mathbf{v}}_{i}-\mathbf{u}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)\right|^{2}$
and integrating over velocities $\tilde{\mathbf{v}}_{i}$ we obtain the
equation

$\displaystyle\displaystyle\frac{\partial T_{i}}{\partial
t}+u_{ix}\frac{\partial
T_{i}}{\partial\tilde{X}_{i}}+\left(u_{iy}-V^{\prime}_{0}tu_{ix}\right)\frac{\partial
T_{i}}{\partial\tilde{Y}_{i}}-\bar{U}_{ix}\left(\tilde{X}_{i},t\right)\frac{\partial
T_{i}}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i},t\right)\frac{\partial
T_{i}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle=-\frac{2}{3}T_{i}\left(\frac{\partial
u_{ix}}{\partial\tilde{X}_{i}}-V^{\prime}_{0}t\frac{\partial
u_{ix}}{\partial\tilde{Y}_{i}}+\frac{\partial
u_{iy}}{\partial\tilde{Y}_{i}}\right),$ (56)

which determines the evolution of the ion temperature in the poloidal sheared
flow with convective flows. The definitions of the ion number density
$n_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$, the velocity
$\mathbf{u}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$ and the ion
temperature $T_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$ are the usual
ones:

$\displaystyle\displaystyle
n_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)=\int
d\tilde{\mathbf{v}}_{i}\bar{F}_{i}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right),$
(57)
$\displaystyle\displaystyle\mathbf{u}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)=\frac{1}{n_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}\int
d\tilde{\mathbf{v}}_{i}\tilde{\mathbf{v}}_{i}\bar{F}_{i},$ (58)
$\displaystyle\displaystyle
T_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)=\frac{m_{i}}{3n_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)}\int
d\tilde{\mathbf{v}}_{i}\left|\tilde{\mathbf{v}}_{i}-\mathbf{u}_{i}\right|^{2}\bar{F}_{i},$
(59)

and $P_{i}=n_{i}T_{i}$ is the ion thermal pressure. Equations (50) - (53) for
ions and the similar equations for electrons, and the Poisson equation for the
potential $\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$,

$\displaystyle\displaystyle\frac{\partial^{2}\Phi_{i}}{\partial^{2}\tilde{X_{i}}}+\frac{\partial^{2}\Phi_{i}}{\partial^{2}\tilde{Y_{i}}}=-4\pi\left(e_{i}n_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)-|e|n^{(i)}_{e}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)\right),$
(60)

where $n^{(i)}_{e}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$ is the electron
density perturbation, determined in the ion frame variables
$\tilde{X}_{i},\tilde{Y}_{i}$, in which potential
$\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},t\right)$ in Eq. (60) is
determined, compose the basic system of the fluid equations, which governs the
mesoscale evolution of the poloidal sheared plasma flow with radially
inhomogeneous convective flows across the magnetic field.

As it follows from Eqs. (95), (96) and (99), (100), the velocities
$\bar{\mathbf{U}}_{i}\left(\tilde{X}_{i},t\right)$,
$\bar{\mathbf{U}}_{e}\left(\tilde{X}_{i},t\right)$ of the ion and electron
convective flows are proportional to the mesoscale gradient of the spectral
intensity $\left|\varphi_{i}\left(\mathbf{k},\tilde{X}_{i}\right)\right|^{2}$
of the turbulent electric field. Therefore the maximum of the ion convective
flow velocities could be in the pedestal, where the maximum gradients of the
plasma parameters are observed, i. e. for the $\tilde{X}_{i}$ values in the
interval $\tilde{X}_{iT}>\tilde{X}_{i}>\tilde{X}_{iB}$, where $\tilde{X}_{iT}$
and $\tilde{X}_{iB}$ are the coordinates of the pedestal top and of the
pedestal bottom, respectively. Outside this interval, i. e. for the plasma
core, $\tilde{X}_{i}>\tilde{X}_{iT}$, and in the far SOL,
$\tilde{X}_{i}<\tilde{X}_{iB}$, the plasma parameters and of FW field are much
more uniform and the convective flow velocities are more slower. Thus, the
problem of the mesoscale evolution of the pedestal plasma involves the problem
of the stability of a plasma with convective flows with spatially
inhomogeneous velocities. As a first step to the solution of the system of
Eqs. (53) - (56), we find the characteristics of the operator

$\displaystyle\displaystyle\frac{D}{Dt}=\frac{\partial}{\partial
t}-\bar{U}_{ix}\left(\tilde{X}_{i},t\right)\frac{\partial}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i},t\right)\frac{\partial}{\partial\tilde{Y}_{i}},$
(61)

which are determined by the system

$\displaystyle\displaystyle
dt=-\frac{d\tilde{X}_{i}}{\bar{U}_{ix}\left(\tilde{X}_{i},t\right)}=-\frac{d\tilde{Y}_{i}}{\bar{U}_{iy}\left(\tilde{X}_{i},t\right)}.$
(62)

For deriving the simplest solution to system (62), which reveals the effects
of the spatial inhomogeneity of the convective flow velocities, we use in Eq.
(62) the expansions

$\displaystyle\displaystyle\bar{U}_{ix}\left(\tilde{X}_{i},t\right)=\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}^{(0)}_{i},t\right)\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right),$
(63)

and

$\displaystyle\displaystyle\bar{U}_{iy}\left(\tilde{X}_{i},t\right)=\bar{U}^{(0)}_{iy}+\bar{U}^{\prime}_{iy}\left(\tilde{X}^{(0)}_{i},t\right)\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)$
(64)

at the vicinity of an arbitrary coordinate $\tilde{X}^{(0)}_{i}$, and consider
the case of the time-independent velocity compressing rate,
$\bar{U}^{\prime}_{ix}=const$, and of the velocity shearing rate,
$\bar{U}^{\prime}_{iy}=const$. The solution to system (62) for the case of the
flows with stationary uniform compressing and shearing rates is simple and has
a form

$\displaystyle\displaystyle\check{X}_{i}=\frac{1}{\bar{U}^{\prime}_{ix}}\left[\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)\right)e^{\bar{U}^{\prime}_{ix}t}-\bar{U}^{(0)}_{ix}\right],$
(65)

and

$\displaystyle\displaystyle\check{Y}_{i}=\tilde{Y}_{i}+\left(\bar{U}^{(0)}_{iy}-\bar{U}^{(0)}_{ix}\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\right)t-\frac{\bar{U}^{\prime}_{iy}}{\left(\bar{U}^{\prime}_{ix}\right)^{2}}\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)\right),$
(66)

$\check{X}_{i}$ and $\check{Y}_{i}$ are the integrals of system (62) with
expansions (63), (64). Note, that at $t=0$
$\check{X}_{i}=\tilde{X}_{i}-\tilde{X}^{(0)}_{i}$.

Now we perform the transformations of variables $\tilde{X}_{i}$,
$\tilde{Y}_{i}$ in Eqs. (53) - (56) to variables $\check{X}_{i}$,
$\check{Y}_{i}$ and derive the following equations:

$\displaystyle\displaystyle\frac{\partial
n_{i}\left(\check{X}_{i},\check{Y}_{i},t\right)}{\partial
t}+e^{\bar{U}^{\prime}_{ix}t}\frac{\partial}{\partial\check{X}_{i}}\left(n_{i}u_{ix}\left(\check{X}_{i},\check{Y}_{i},t\right)\right)$
$\displaystyle\displaystyle-\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial}{\partial\check{Y}_{i}}\left(n_{i}u_{ix}\left(\check{X}_{i},\check{Y}_{i},t\right)\right)+\frac{\partial}{\partial\check{Y}_{i}}\left(n_{i}\left(u_{iy}\left(\check{X}_{i},\check{Y}_{i},t\right)-V^{\prime}_{0}tu_{ix}\right)\right)=0,$
(67) $\displaystyle\displaystyle\frac{\partial u_{ix}}{\partial
t}+u_{ix}\left(e^{\bar{U}^{\prime}_{ix}t}\frac{\partial
u_{ix}}{\partial\check{X}_{i}}-\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial
u_{ix}}{\partial\check{Y}_{i}}\right)+\left(u_{iy}-V^{\prime}_{0}tu_{ix}\right)\frac{\partial
u_{ix}}{\partial\check{Y}_{i}}$
$\displaystyle\displaystyle=-\frac{1}{m_{i}n_{i}\left(\check{X}_{i},\check{Y}_{i},t\right)}\left(e^{\bar{U}^{\prime}_{ix}t}\frac{\partial
P_{i}}{\partial\check{X}_{i}}-\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial
P_{i}}{\partial\check{Y}_{i}}-V^{\prime}_{0}t\frac{\partial
P_{i}}{\partial\check{Y}_{i}}\right)$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\left(e^{\bar{U}^{\prime}_{ix}t}\frac{\partial}{\partial\check{X}_{i}}\Phi\left(\check{X}_{i},\check{Y}_{i},t\right)-\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial}{\partial\check{Y}_{i}}\Phi\left(\check{X}_{i},\check{Y}_{i},t\right)-V^{\prime}_{0}t\frac{\partial}{\partial\check{Y}_{i}}\Phi\left(\check{X}_{i},\check{Y}_{i},t\right)\right)+\omega_{ci}u_{iy},$
(68)

$\displaystyle\displaystyle\frac{\partial u_{iy}}{\partial
t}+u_{ix}\left(e^{\bar{U}^{\prime}_{ix}t}\frac{\partial
u_{iy}}{\partial\check{X}_{i}}-\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial
u_{iy}}{\partial\check{Y}_{i}}\right)+\left(u_{iy}-V^{\prime}_{0}tu_{ix}\right)\frac{\partial
u_{iy}}{\partial\check{Y}_{i}}$
$\displaystyle\displaystyle=-\frac{1}{m_{i}n_{i}\left(\check{X}_{i},\check{Y}_{i},t\right)}\frac{\partial
P_{i}}{\partial\check{Y}_{i}}-\frac{e_{i}}{m_{i}}\frac{\partial}{\partial\check{Y}_{i}}\Phi\left(\check{X}_{i},\check{Y}_{i},t\right)-\left(\omega_{ci}+V^{\prime}_{0}\right)u_{ix},$
(69) $\displaystyle\displaystyle\frac{\partial T_{i}}{\partial
t}+u_{ix}\left(e^{\bar{U}^{\prime}_{ix}t}\frac{\partial
T_{i}}{\partial\check{X}_{i}}-\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial
T_{i}}{\partial\check{Y}_{i}}\right)+\left(u_{iy}-V^{\prime}_{0}tu_{ix}\right)\frac{\partial
T_{i}}{\partial\check{Y}_{i}}$
$\displaystyle\displaystyle=-\frac{2}{3}T_{i}\left(e^{\bar{U}^{\prime}_{ix}t}\frac{\partial
u_{ix}}{\partial\check{X}_{i}}-\frac{U^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial
u_{ix}}{\partial\check{Y}_{i}}+\frac{\partial
u_{iy}}{\partial\check{Y}_{i}}\right),$ (70)

$\displaystyle\displaystyle\left(e^{2\bar{U}^{\prime}_{ix}t}\frac{\partial^{2}}{\partial\check{X}^{2}_{i}}-2e^{\bar{U}^{\prime}_{ix}t}\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\frac{\partial^{2}}{\partial\check{X}_{i}\partial\check{Y}_{i}}+\left(\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\right)^{2}\frac{\partial^{2}}{\partial\check{Y}_{i}^{2}}\right)\Phi\left(\check{X}_{i},\check{Y}_{i},t\right)$
$\displaystyle\displaystyle=-4\pi\left(e_{i}n_{i}\left(\check{X}_{i},\check{Y}_{i},t\right)-|e|n^{(i)}_{e}\left(\check{X}_{i},\check{Y}_{i},t\right)\right).$
(71)

This system does not contain any explicit dependences on the radial coordinate
$\check{X}_{i}$. Instead, these equations involve the explicit time
dependences, which reveal the effects of the basic poloidal sheared flow and
of the convective compressed and sheared flows on the mesoscale temporal
evolution of the inhomogeneous turbulent plasmas. It follows from this system,
that the basic poloidal sheared flow with the velocity shearing rate
$V^{\prime}_{0}$ leads to the appearance of the linearly growing with time
coefficients in Eqs. (67) - (71). This time dependence reveals the effect of
the continuous distortion with time of the perturbations by the basic poloidal
sheared flow. This distortion grows with time and forms a time-dependent
nonmodal processes, the linear and the renormalized nonlinear stages of which
were investigated in Refs. Mikhailenko1 ; Mikhailenko2 ; Mikhailenko3 . It was
foundMikhailenko1 ; Mikhailenko2 ; Mikhailenko3 , that this nonmodal process
is at the foundation of the experimentally observed effect of the suppression
of the drift-type instabilities by the poloidal sheared flow, which have the
growth rates less than the poloidal flow velocity shearing rate
$V^{\prime}_{0}$. Equations (67) - (71) display also principally different
effect of the compressed convective flow along $\check{X}_{i}$. It reveals in
the exponentially growing with time coefficients $e^{\bar{U}^{\prime}_{ix}t}$
in Eqs. (67) - (71), which at time $t>\left(\bar{U}^{\prime}_{ix}\right)^{-1}$
determine the compressed flow as the dominant factor in the evolution of the
plasma with a radially inhomogeneous turbulence. For that time, system (67) -
(71) for $n_{i},n_{e}$, $T_{i},T_{e}$, $u_{ix},u_{ex}$, $u_{iy},u_{ey}$, and
$\Phi$ can be substantially reduced by neglecting the derivatives over
$\check{Y}_{i}$, which are exponentially small with respect to the terms
containing the derivatives over $\check{X}_{i}$. The reduced system,

$\displaystyle\displaystyle\frac{\partial
n_{i}\left(\check{X}_{i},t\right)}{\partial
t}+e^{\bar{U}^{\prime}_{ix}t}\frac{\partial}{\partial\check{X}_{i}}\left(n_{i}\left(\check{X}_{i}\right)u_{ix}\left(\check{X}_{i},t\right)\right)=0,$
(72) $\displaystyle\displaystyle\frac{\partial
u_{ix}\left(\check{X}_{i},t\right)}{\partial
t}+e^{\bar{U}^{\prime}_{ix}t}u_{ix}\frac{\partial
u_{ix}}{\partial\check{X}_{i}}$
$\displaystyle\displaystyle=-\frac{e^{\bar{U}^{\prime}_{ix}t}}{m_{i}}\left(\frac{1}{n_{i}\left(\check{X}_{i}\right)}\frac{\partial
P_{i}}{\partial\check{X}_{i}}-e_{i}\frac{\partial\Phi\left(\check{X}_{i},t\right)}{\partial\check{X}_{i}}\right)+\omega_{ci}u_{iy},$
(73) $\displaystyle\displaystyle\frac{\partial
u_{iy}\left(\check{X}_{i},t\right)}{\partial
t}+e^{\bar{U}^{\prime}_{ix}t}u_{ix}\frac{\partial
u_{iy}}{\partial\check{X}_{i}}=-\omega_{ci}u_{ix},$ (74)
$\displaystyle\displaystyle\frac{\partial
T_{i}\left(\check{X}_{i},t\right)}{\partial
t}+e^{\bar{U}^{\prime}_{ix}t}u_{ix}\frac{\partial
T_{i}}{\partial\check{X}_{i}}=-e^{\bar{U}^{\prime}_{ix}t}\frac{2}{3}T_{i}\frac{\partial
u_{ix}}{\partial\check{X}_{i}},$ (75)

and the equations for $n_{e}$, $T_{e}$, $u_{ex}$, $u_{ey}$ derived from Eqs.
(72) - (75) by changing species index $i$ on $e$, supplemented by the equation
for the electrostatic potential $\Phi\left(\check{X}_{i},t\right)$,

$\displaystyle\displaystyle
e^{2\bar{U}^{\prime}_{ix}t}\frac{\partial^{2}\Phi\left(\check{X}_{i},t\right)}{\partial\check{X}_{i}^{2}}=-4\pi\left(e_{i}n_{i}\left(\check{X}_{i},t\right)-|e|n^{(i)}_{e}\left(\check{X}_{i},t\right)\right),$
(76)

compose the nonlinear system of equations, which governs the mesoscale
dynamics of the pedestal plasma. The more simple presentation of all these
equations may be derived using the ion ”compressed” time variable $\tau_{i}$,
determined by the relation

$\displaystyle\displaystyle\bar{U}_{ix}^{\prime}\tau_{i}=e^{\bar{U}_{ix}^{\prime}t},$
(77)

with which Eqs. (72) - (76) are presented in the following much more simple
form:

$\displaystyle\displaystyle\frac{\partial
n_{i}\left(\check{X}_{i},\tau_{i}\right)}{\partial\tau_{i}}+\frac{\partial}{\partial\check{X}_{i}}\left(n_{i}u_{ix}\left(\check{X}_{i},\tau_{i}\right)\right)=0,$
(78) $\displaystyle\displaystyle\frac{\partial
u_{ix}\left(\check{X}_{i},\tau_{i}\right)}{\partial\tau_{i}}+u_{ix}\frac{\partial
u_{ix}}{\partial\check{X}_{i}}$
$\displaystyle\displaystyle=-\frac{1}{m_{i}}\left(\frac{1}{n_{i}\left(\check{X}_{i},\tau_{i}\right)}\frac{\partial
P_{i}}{\partial\check{X}_{i}}-e_{i}\frac{\partial\Phi\left(\check{X}_{i},\tau_{i}\right)}{\partial\check{X}_{i}}\right)+\frac{\omega_{ci}}{\bar{U}_{ix}^{\prime}\tau}u_{iy}\left(\check{X}_{i},\tau_{i}\right),$
(79) $\displaystyle\displaystyle\frac{\partial
u_{iy}\left(\check{X}_{i},\tau_{i}\right)}{\partial\tau_{i}}+u_{ix}\frac{\partial
u_{iy}}{\partial\check{X}_{i}}=-\frac{\omega_{ci}}{\bar{U}_{ix}^{\prime}\tau_{i}}u_{ix}\left(\check{X}_{i},\tau_{i}\right),$
(80) $\displaystyle\displaystyle\frac{\partial
T_{i}\left(\check{X}_{i},\tau_{i}\right)}{\partial\tau_{i}}+u_{ix}\frac{\partial
T_{i}}{\partial\check{X}_{i}}+T_{i}\frac{\partial
u_{ix}\left(\check{X}_{i},\tau_{i}\right)}{\partial\check{X}_{i}}=0.$ (81)

Using Eqs. (77) - (81) with species index $i$ changed on $e$, we derive the
equations for $n_{e}\left(\check{X}_{e},\tau_{e}\right)$,
$T_{e}\left(\check{X}_{e},\tau_{e}\right)$,
$u_{ex}\left(\check{X}_{e},\tau_{e}\right)$,
$u_{ey}\left(\check{X}_{e},\tau_{e}\right)$. Also, Eq. (76) for the
electrostatic potential $\Phi\left(\check{X}_{i},\tau_{i}\right)$, will get a
form

$\displaystyle\displaystyle\frac{\partial^{2}\Phi_{i}\left(\check{X}_{i},\tau_{i}\right)}{\partial\check{X}_{i}^{2}}=$
$\displaystyle\displaystyle=-\frac{4\pi}{\left(\bar{U}_{ix}^{\prime}\tau_{i}\right)^{2}}\left(e_{i}n_{i}\left(\check{X}_{i},\tau_{i}\right)-|e|n^{(i)}_{e}\left(\check{X}_{i},\tau_{i}\right)\right).$
(82)

For understanding the temporal evolution of the plasma compressing flow it is
instructive to derive the linearised solutions to system (78) - (82). We
present the ion density and the ion temperature in Eqs. (78) and (81) in the
forms

$\displaystyle\displaystyle
n_{i}\left(\check{X}_{i},\tau_{i}\right)=n_{i0}\left(\check{X}_{i},\tau_{i}\right)+n_{i1}\left(\check{X}_{i},\tau_{i}\right),$
(83)

and

$\displaystyle\displaystyle
T_{i}\left(\check{X}_{i},\tau_{i}\right)=T_{i0}\left(\check{X}_{i},\tau_{i}\right)+T_{i1}\left(\check{X}_{i},\tau_{i}\right),$
(84)

where $n_{i0}\left(\check{X}_{i},\tau_{i}\right)$ and
$T_{i0}\left(\check{X}_{i},\tau_{i}\right)$ are the equilibrium values of the
ion density and of the ion temperature, and $n_{i1}$, $T_{i1}$ are their
perturbations caused by the self-consistent electrostatic respond of a plasma
on the relative motion of the ion and electron compressed flows. As it follows
from Eqs. (78) and (81), the evolution with time of the ion density
$n_{i0}\left(\check{X}_{i},\tau_{i}\right)$ and of the ion temperature
$T_{i0}\left(\check{X}_{i},\tau_{i}\right)$ in compressed flow is determined
by the equations

$\displaystyle\displaystyle\frac{\partial
n_{i0}\left(\check{X}_{i},\tau_{i}\right)}{\partial\tau_{i}}=0,$ (85)
$\displaystyle\displaystyle\frac{\partial
T_{i0}\left(\check{X}_{i},\tau_{i}\right)}{\partial\tau_{i}}=0,$ (86)

or, for $n_{i0}\left(\tilde{X}_{i},\tau_{i}\right)$ and
$T_{i0}\left(\tilde{X}_{i},\tau_{i}\right)$, by the equations

$\displaystyle\displaystyle\frac{\partial n_{i0}}{\partial
t}-\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)\right)\frac{\partial
n_{i0}}{\partial\tilde{X}_{i}}=0,$ (87)
$\displaystyle\displaystyle\frac{\partial T_{i0}}{\partial
t}-\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)\right)\frac{\partial
T_{i0}}{\partial\tilde{X}_{i}}=0,$ (88)

where expansion (63) was used. The solution to Eq. (85),

$\displaystyle\displaystyle n_{i0}=n_{i0}\left(\check{X}_{i}\right),$ (89)

is independent on time, whereas the solution
$n_{i0}\left(\tilde{X}_{i},t\right)$ to Eq. (87) for the simplest initial
condition

$\displaystyle\displaystyle
n_{i0}\left(\tilde{X}_{i},t=0\right)=n_{i0}\left(\check{X}_{i}\right)=n_{i0}\left(\tilde{X}_{i0}\right)+\frac{\partial
n_{i0}\left(\tilde{X}_{i},t=0\right)}{\partial\tilde{X}_{i}}|_{\tilde{X}_{i}=\tilde{X}_{i0}}\check{X}_{i}$
(90)

is equal to

$\displaystyle\displaystyle
n_{i0}\left(\tilde{X}_{i},t\right)=n_{i0}\left(\tilde{X}_{i0}\right)+\frac{\partial
n_{i0}\left(\tilde{X}_{i},t=0\right)}{\partial\tilde{X}_{i}}|_{\tilde{X}_{i}=\tilde{X}_{i0}}$
$\displaystyle\displaystyle\times\frac{1}{\bar{U}^{\prime}_{ix}}\left[\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)\right)e^{\bar{U}^{\prime}_{ix}t}-\bar{U}^{(0)}_{ix}\right].$
(91)

Equation (91) displays that the value of $n_{i0}$ at
$\tilde{X}_{i}=\tilde{X}_{i1}$, detected at $t=t_{1}$, is shifted by the
compressed flow and is observed in
$\tilde{X}_{i}=\tilde{X}_{i2}<\tilde{X}_{i1}$ at time $t=t_{2}>t_{1}$. The
solution similar to Eq. (91) is derived also to Eq. (88) for the temporal
evolution of the equilibrium ion temperature $T_{i0}$ in compressed flow.

In the limit of the vanished compressing rate $\bar{U}^{\prime}_{ix}$, Eq.
(91) displays the transport of the ion density inhomogeneity with velocity
$\bar{U}_{ix}\left(\tilde{X}_{i0}\right)$,

$\displaystyle\displaystyle
n_{i0}\left(\tilde{X}_{i},t\right)=n_{i0}\left(\tilde{X}_{i0}\right)+\frac{\partial
n_{i0}\left(\tilde{X}_{i},t=0\right)}{\partial\tilde{X}_{i}}|_{\tilde{X}_{i}=\tilde{X}_{i0}}\cdot\left(\tilde{X}_{i}-\tilde{X}_{i0}+\bar{U}^{(0)}_{ix}t\right).$
(92)

When the level of IC parametric turbulence is much lover than the level of the
low frequency drift turbulence, the electric field (16) is determined by the
drift turbulence. The suppression of the edge drift turbulence by the poloidal
sheared flow entails the formation of the stagnation point for the compressed
flow velocity at the pedestal bottom, where
$\bar{U}_{ix}\left(\tilde{X}_{iB}\right)\approx 0$ and
$n_{i0}\left(\tilde{X}_{i}<\tilde{X}_{iB}\right)\approx 0$. The solution (91)
for the ion density in the region $\tilde{X}_{i}>\tilde{X}_{iB}$ of the
pedestal bottom becomes equal to

$\displaystyle\displaystyle n_{i0}\left(\tilde{X}_{i},t\right)=\frac{\partial
n_{i0}\left(\tilde{X}_{i}\right)}{\partial\tilde{X}_{i}}|_{\tilde{X}_{i}=\tilde{X}_{iB}}e^{\bar{U}^{\prime}_{ix}\left(\tilde{X}_{iB}\right)t}\left(\tilde{X}_{i}-\tilde{X}_{iB}\right).$
(93)

It follows from Eqs. (93), that the gradient of the ion density at
$\tilde{X}_{i}>\tilde{X}_{iB}$ grows exponentially with time as
$e^{\bar{U}^{\prime}_{ix}t}$. This effect of the fast stepping up with time of
the density profile in the pedestal region by the compressed flow looks like
the instability development with the growth rate equal to
$\bar{U}^{\prime}_{ix}$ for ions and $\bar{U}^{\prime}_{ex}$ for electrons.

It follows from Eq. (73) that due to the fast growing coefficient
$e^{\bar{U}^{\prime}_{ix}t}$ the radial ion pressure force at some time
$t\gtrsim t_{\star}$ can be larger than the radial component of the ion
Lorentz force. The crude estimate for the transition time $t_{\star}$ may be
derived from the balance relation, which follows from Eq. (73),

$\displaystyle\displaystyle
e^{\bar{U}^{\prime}_{ix}t_{\star}}\frac{v_{Ti}}{L_{n}}\sim\omega_{ci},$ (94)

where $L_{n}$ is the spatial scale of the ion density gradient of the pedestal
plasma. For the ion temperature $T_{i}\sim 200$ eV, $L_{n}=2$ cm, and
$B_{0}=2$ T, we have $e^{\bar{U}^{\prime}_{ix}t_{\star}}\sim 10$. For the
numerical sample, considered in Ref. Mikhailenko , $\bar{U}_{ix}\sim
0.1v_{Ti}\sim 10^{6}$ cm/s and $\bar{U}^{\prime}_{ix}\sim 5\cdot 10^{5}$ s-1
we derive $t_{\star}\sim 6\cdot 10^{-6}$ s. At time $t>t_{\star}$, the radial
outflow of the temporally unconfined ions forms.

##  V Conclusions

In this paper, we develop the kinetic and hydrodynamic theories of the
convective mesoscale flows, driven by the spatially inhomogeneous IC
parametric microturbulence of the pedestal plasma with a sheared poloidal
flow. The amplitude $\tilde{\mathbf{E}}\left(\mathbf{k},X_{i}\right)$ of the
electric field (16) of the microscale IC parametric turbulence of the
inhomogeneous plasma, driven by the inhomogeneous FW field, is spatially
inhomogeneous and depends on the mesoscale coordinate $X_{i}$. In this paper,
the IC microturbulence is assumed to be weakly nonlinear with known dependence
of $\tilde{\mathbf{E}}$ on $X_{i}$, which established by the linear theory of
the IC instability, and by the weak nonlinear theory which determines the
local saturation this instability at position $X_{i}$. The basic result of the
developed theory is the Vlasov equation (41) which determines the mesoscale
evolution of the ion/electron distribution functions resulted from the
interaction of ions/electrons with the inhomogeneous IC microturbulence. This
theory predicts the generation of the sheared poloidal convective flow, and of
the radial compressed flow with radial flow velocity gradient.

The developed hydrodynamic mesoscale theory 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. This
gradient density growth is limited by the radial oscillating with time outflow
of the pedestal ions to SOL. The process of the temporal steeping and
smoothing of the plasma density profile in the pedestal reveals as the
manifestation of the ultimate stage of the relaxation of the unstable radially
inhomogeneous density of the pedestal plasma. It was initiated as the
microturbulence and finalised as the mesoscale nonlinear relaxation
oscillations. The analytical treatment of the temporal evolution of the plasma
compressed flow near the pedestal bottom at time $t>t_{\star}$ may be
performed as the solution of the initial-boundary problem for the system of
equations which includes Eqs. (81) - (84) for
$n_{i1}\left(\check{X}_{i},\tau_{i}\right)$,
$T_{i1}\left(\check{X}_{i},\tau_{i}\right)$,
$u_{ix}\left(\check{X}_{i},\tau_{i}\right)$,
$u_{iy}\left(\check{X}_{i},\tau_{i}\right)$, Eqs. (81) - (84) for the electron
component for $n_{e1}\left(\check{X}_{e},\tau_{e}\right)$,
$T_{e1}\left(\check{X}_{e},\tau_{e}\right)$,
$u_{ex}\left(\check{X}_{e},\tau_{e}\right)$,
$u_{ey}\left(\check{X}_{e},\tau_{e}\right)$, and Eq. (85) for potential
$\Phi_{i}\left(\check{X}_{i},\tau_{i}\right)$. The solution of this problem
will be done in the separate paper.

###### 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 $\bar{U}^{(0)}_{ix}\left(\tilde{X}_{i},t\right)$ and
$\bar{U}^{(0)}_{iy}\left(\tilde{X}_{i},t\right)$

For the electric field $\tilde{\mathbf{E}}_{i}$, determined by Eq. (16),
velocities $\bar{U}^{(0)}_{ix}\left(\tilde{X}_{i},t\right)$ and
$\bar{U}^{(0)}_{iy}\left(\tilde{X}_{i},t\right)$ were determined in
Ref.Mikhailenko . For completeness, we present these results here, i. e.

$\displaystyle\displaystyle\bar{U}^{(0)}_{ix}\left(\tilde{X}_{i}\right)=\frac{1}{2\omega_{ci}}\frac{e_{i}^{2}}{m^{2}_{i}}\int
d\mathbf{k}\left[a_{i1}\left(\mathbf{k}\right)\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i}\right)\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}^{\ast}_{iy}\left(\mathbf{k},\tilde{X}_{i}\right)\right)\right.$
$\displaystyle\displaystyle\left.+a_{i2}\left(\mathbf{k}\right)\tilde{E}_{iy}\left(\mathbf{k},\tilde{X}_{i}\right)\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}^{\ast}_{ix}\left(\mathbf{k},\tilde{X}_{i}\right)\right)\right]$
$\displaystyle\displaystyle=\frac{1}{4\omega_{ci}}\frac{e_{i}^{2}}{m^{2}_{i}}\int
d\mathbf{k}k_{x}k_{y}\left(a_{i1}\left(\mathbf{k}\right)+a_{i2}\left(\mathbf{k}\right)\right)\frac{\partial}{\partial\tilde{X}_{i}}\left|\varphi\left(\mathbf{k},\tilde{X}_{i}\right)\right|^{2},$
(95)

and

$\displaystyle\displaystyle\bar{U}^{(0)}_{iy}\left(\tilde{X}_{i}\right)=-\frac{1}{4\omega_{ci}}\frac{e_{i}^{2}}{m^{2}_{i}}\int
d\mathbf{k}\left[a_{i1}\left(\mathbf{k}\right)\frac{\partial}{\partial\tilde{X}_{i}}\left|\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i}\right)\right|^{2}-a_{i2}\left(\mathbf{k}\right)\frac{\partial}{\partial
X_{i}}\left|\tilde{E}_{iy}\left(\mathbf{k},\tilde{X}_{i}\right)\right|^{2}\right]$
$\displaystyle\displaystyle=-\frac{1}{4\omega_{ci}}\frac{e_{i}^{2}}{m^{2}_{i}}\int
d\mathbf{k}\left(k^{2}_{x}a_{i1}\left(\mathbf{k}\right)-k^{2}_{y}a_{i2}\left(\mathbf{k}\right)\right)\frac{\partial}{\partial\tilde{X}_{i}}\left|\varphi\left(\mathbf{k},\tilde{X}_{i}\right)\right|^{2},$
(96)

where the asterisk in Eq. (95) implies the operation of complex conjugate. The
coefficients $a_{i1}\left(\mathbf{k}\right)$ and
$a_{i2}\left(\mathbf{k}\right)$ are determined as

$\displaystyle\displaystyle
a_{i1}\left(\mathbf{k}\right)=\left[\frac{\omega_{ci}}{\omega\left(\mathbf{k}\right)\left(\omega_{ci}+\omega\left(\mathbf{k}\right)\right)^{2}}+\frac{\omega_{ci}}{\omega\left(\mathbf{k}\right)\left(\omega_{ci}-\omega\left(\mathbf{k}\right)\right)^{2}}+\frac{1}{\left(\omega^{2}_{ci}-\omega^{2}\left(\mathbf{k}\right)\right)}\right],$
(97)

and

$\displaystyle\displaystyle
a_{i2}\left(\mathbf{k}\right)=\left[\frac{1}{\left(\omega_{ci}+\omega\left(\mathbf{k}\right)\right)^{2}}+\frac{1}{\left(\omega_{ci}-\omega\left(\mathbf{k}\right)\right)^{2}}+\frac{1}{\left(\omega^{2}_{ci}-\omega^{2}\left(\mathbf{k}\right)\right)}\right].$
(98)

The velocities of electrons $\bar{U}^{(0)}_{ex}\left(\tilde{X}_{i}\right)$ and
$\bar{U}^{(0)}_{ey}\left(\tilde{X}_{i}\right)$ in the ion frame, with electric
field $\mathbf{E}_{e}\left(\mathbf{\hat{r}}_{i},\tilde{X}_{i},t\right)$
determined by Eq. (17), are

$\displaystyle\displaystyle\bar{U}^{(0)}_{ex}\left(\tilde{X}_{i}\right)\approx\frac{c^{2}}{B^{2}_{0}}\int
d\mathbf{k}\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i}\right)\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}^{\ast}_{iy}\left(\mathbf{k},\tilde{X}_{i}\right)\right)$
$\displaystyle\displaystyle\times\sum\limits_{p=-\infty}^{\infty}J^{2}_{p}\left(a_{ie}\left(\mathbf{k},\tilde{X}_{i}\right)\right)\frac{1}{\Omega_{p}\left(\mathbf{k},\tilde{X}_{i}\right)},$
(99)

and

$\displaystyle\displaystyle\bar{U}^{(0)}_{ey}\left(\tilde{X}_{i}\right)\approx-\frac{1}{2}\frac{c^{2}}{B^{2}_{0}}\int
d\mathbf{k}\frac{\partial}{\partial\tilde{X}_{i}}\left|\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i}\right)\right|^{2}$
$\displaystyle\displaystyle\times\sum\limits_{p=-\infty}^{\infty}J^{2}_{p}\left(a_{ie}\left(\mathbf{k},\tilde{X}_{i}\right)\right)\frac{1}{\Omega_{p}\left(\mathbf{k},\tilde{X}_{i}\right)},$
(100)

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

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