# Transport Barrier and Spinning Blob Dynamics in the Tokamak Edge

Junyi Cheng  Renewable and Sustainable Energy Institute, University of
Colorado, Boulder, Colorado 80309, USA  James Myra  Lodestar Research
Corporation, 360 Bobolink Court, Louisville, Colorado 80027, USA  Seung-Hoe Ku
Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA  Robert
Hager  Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA
Choong-Seock Chang  Princeton Plasma Physics Laboratory, Princeton, New Jersey
08543, USA  Scott Parker  Renewable and Sustainable Energy Institute,
University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

In this work, we investigate the dynamics of plasma blobs in the edge of
magnetic confinement devices using a full-f gyrokinetic particle-in-cell code
with X-point geometry. In simulations, the evolution of a seeded blob is
followed as it approaches a naturally-forming zonal shear layer near the
separatrix, where the blob is stabilized by a large spin induced by the self-
consistent adiabatic electron response, and blob bifurcation and trapping are
observed during the cross-field propagation of blobs. A new theoretical
explanation in both the zonal free and zonal shear layer is constructed, where
the dominant $\mathbf{E}\times\mathbf{B}$ spin motion is included. A
theoretical condition for a transport barrier induced by the interaction
between spinning blobs and the zonal shear layer is obtained, and its scaling
is verified with simulations. The new theoretical framework, especially the
transport barrier, is applicable to explain and predict various experimental
phenomena. In particular, the transport barrier condition calculated with
experimental parameters demonstrates that the blob radial transport for H mode
is smaller than L mode in experiments.

Coherent structures having poloidally and radially localized density
structures and extended along the magnetic field, known as plasma blobs, often
dominate the cross-field transport of particles and energy in the edge region
between the confined plasma and material surface in magnetic confinement
devices. The cross-field propagation of plasma blobs is of intrinsic interest
for a better understanding of edge transport in future toroidal fusion devices
and has drawn significant attention from the physics community [1, 2, 3, 4, 5,
6, 7, 8, 9, 10]. Furthermore, we will see that rapidly spinning blobs,
considered here, have vortex interactions with a sheared flow layer that are
of fundamental interest, reminiscent of similar phenomena in 2D neutral
fluids.

In the conventional paradigm[1] of convective blob transport, the charge
separation induced by opposite ion and electron vertical drift leads to a
dipole potential and such a potential leads to an outward
$\mathbf{E}\times\mathbf{B}$ drift. Although much progress has been made in
understanding the nonlinear dynamics of these structures [2, 3], realistic
experimental conditions are complicated and further investigation is
warranted. For example, the mushroom shape of blobs [11] due to charge
separation has been demonstrated in a linear plasma device; however, blobs in
the tokamak have a more stable nearly circular cross-section[12, 13] shape.
Blobs with negative cross-field velocity (radial velocity in a tokamak) have
been observed [12, 14]. The blob emission in the SOL (scrape-off layer) region
in H mode is less than L mode, which is observed in different tokamaks, such
as NSTX[15] and DIII-D[13].

In this work, we investigate the blob dynamics by carrying out simulations
using the edge-physics-aimed 5D particle-in-cell (PIC) gyrokinetic code XGC
[16, 17]. The blobs are seeded inside the separatrix using a realistic
magnetic equilibrium from a C-Mod H-mode discharge[18]. We consider the
interaction of blobs born just inside the separatrix with a naturally forming
shear layer near the separatrix. In the simulations, blobs with a large
amplitude are stable with large spin and small cross-field motion. It is found
that under certain conditions the spinning blob bifurcates, or splits, into
two distinct parts at the midplane where the inner part is trapped in the core
region, and the outer part moves outward radially into the SOL region. Hence,
the $E_{r}$ shear layer near the separatrix acts as a transport barrier. A new
theory that includes the dominant $\mathbf{E}\times\mathbf{B}$ spin motion in
the parallel Ohm’s law, vorticity and continuity equations is required. The
potential coming from the adiabatic electron response giving rise to the spin
is treated as zero-order, whereas the dipole potential is first-order. The
blob can be stabilized by spinning and distorted by the large $E_{r}$ shear
near the separatrix. The blob’s radial motion responds to the zonal
electrostatic potential well with the blob being charged positive, consistent
with the simulation results. The new theory may explain the negative radial
velocity of blobs seen in experiments. The blob bifurcation in the separatrix
region is also observed in experiments, and the transport barrier condition
obtained here and calculated with experimental parameters demonstrates that
the blob radial transport for H mode is smaller than L mode in experiments.

Simulation results. Here, the electrostatic version [19, 17] of XGC is used
for simulations, which employs drift kinetic electrons and gyrokinetic ions
with a logical sheath boundary condition [20] at the divertor plate. For
simplicity, the source term is turned off, and no neutral-plasma interaction
is included. The geometry and magnetic equilibrium are based on a lower single
null Alcator C-Mod EDA high (H) mode discharge [18] $\\#1160930033$ with a
plasma current of $I_{p}=1.4\mathrm{MA}$, a toroidal magnetic field on the
axis of $5.8\mathrm{T}$, an input power of $5.4\mathrm{MW}$, and a record
volume averaged pressure exceeding $0.2\mathrm{MPa}$. We simplify the problem
in two ways. First, we set the density and temperature profiles to be uniform
to avoid complications of gradient-driven instabilities super-imposed on the
seeded blob physics. Second, in order to reduce the required computational
resources, we reduce the magnetic field by a factor of $8$, resulting in a
magnetic field magnitude of $0.6\mathrm{T}$ at the separatrix at the outer-
most midplane. (Experimental conditions will be restored in our final scaling
estimates.) The ion grad-$B$ drift is directed downwards. The ions and
electrons have a uniform temperature profile $T_{e}=T_{i}=60\mathrm{eV}$ and
the density profile is uniform $n_{e}=n_{i}=10^{19}\mathrm{m^{-3}}$. Note,
since the gradients of blob density and the profile satisfy
$\nabla\mathrm{ln}n_{b}>\nabla\mathrm{ln}n_{0}$, the turbulence generated by
profile gradients is suppressed. Thus simulations here are used only to
investigate the blob physics.

First, we carry out a simulation without blobs as a controlled case. No
turbulence is observed. A small amplitude zonal potential develops at the
separatrix and the flux surface at the “virtual” upper X-point due to the
charge absorption at the divertor. This controlled simulation verifies
turbulence is suppressed with uniform density and temperature profiles.

Figure 1: The blob structure of the electrostatic potential $\phi$ in the $RZ$
plane at different times. The left line indicates the flux surface where the
blob is initialized and the right line indicates the separatrix surface. For
instance, at $t=28.02\mathrm{\mu s}$, the blob radial velocity
$v_{b}^{R}\approx 76.16\mathrm{m/s}$, the blob spin velocity $v_{s}\approx
11565.12\mathrm{m/s}$, the averaged blob perturbed density $\delta n$ is about
$0.51$ times of background density, and the adiabaticity $e\phi/T/(\delta
n/n)\lesssim 1$.  Figure 2: The blob center-of-mass radial velocity
$v_{b}^{R}$ versus time compared to the averaged radial component of
$\mathbf{E}\times\mathbf{B}$ drift $\langle v_{E}^{R}\rangle$ versus time.
Averaging $\langle...\rangle$ is done over the blob electron density.

We now carry out a simulation where a blob is seeded in the closed-flux region
just inside the separatrix at $\psi/\psi_{x}=0.94$, where $\psi$ is the
poloidal flux and $\psi_{x}$ is the poloidal flux at the separatrix. Our
simulations are not intended to model the main pedestal where Reynolds stress
and plasma profile gradients play an important role. The radial size of the
seeded blobs is $3.71$ times the ion gyro radius. The structure of the seeded
blob is Gaussian in the radial coordinate and the other bi-normal coordinate
and a cosine function along the field line. The amplitude of density
perturbation is an input parameter. When the density perturbation is small,
the blob has an obvious charge separation and a radial outward motion; when
the density perturbation is large enough, as shown in Fig. 1, the blob
structure is stable due to a fast spin induced by the adiabatic electron
response. The blob moves radially outward in the closed-flux region at a speed
much smaller than the spin velocity. This behavior is expected from previous
fluid studies in the SOL in simplified geometry[21, 22]. There is also a small
downward drift until the blob’s radial motion stops. Then the blob bifurcates
into two: one blob stays inside the separatrix and continues to drift
downward; the other smaller blob moves across the separatrix. Hence, the
separatrix region acts as a transport barrier for the inner blob.

The blob radial velocity $v_{b}^{R}$ can be measured by taking the time
derivative of the center-of-mass[23] location at each time step. The radial
component of the $\mathbf{E}\times\mathbf{B}$ drift averaged over the blob
electron density, $\langle v_{E}^{R}\rangle$, indicates that the radial motion
is caused by the average $\mathbf{E}\times\mathbf{B}$ drift. As shown in Fig.
2, $v_{b}^{R}$ and $\langle v_{E}^{R}\rangle$ are in reasonable agreement. The
dipole potential structure in the vertical direction must still exist since it
causes a pure radial $\mathbf{E}\times\mathbf{B}$ radial motion. However, it
is not visible in Fig. 1 because the adiabatic potential dominates. In what
follows, the dipole potential structure will be treated as a first-order
expansion of the zero-order positive potential due to the adiabatic electron
response.

Theoretical explanation of results. We start with the usual fluid vorticity
equation

$\frac{\omega_{pi}^{2}}{4\pi\Omega_{i}^{2}}\left(\frac{\partial}{\partial
t}+\mathbf{v}_{E}\right)\nabla^{2}_{\perp}\phi=\nabla
J_{\|}+\frac{2}{B}\mathbb{b}\times\kappa\cdot\nabla p,$ (1)

and the parallel Ohm’s law

$\nu J_{\|}+\mathbf{v}_{E}\cdot\nabla
J_{\|}=\frac{ne^{2}}{m_{e}}\left(\frac{T_{e}}{e}\nabla_{\|}\mathrm{In}n-\nabla\phi\right),$
(2)

along with the fluid continuity equation, which is used to eliminate density
and get $J_{\parallel}(\phi)$. Note that the spin induced by
$\mathbf{E}\times\mathbf{B}$ drift dominates the blob motion, thus the
parallel Ohm’s law Eq. 2 and the fluid equations include the
$\mathbf{E}\times\mathbf{B}$ and spin motion in leading order. Here
$\omega_{pi}$ is the ion plasma frequency, $\Omega_{i}$ is the ion
gyrofrequency, $\mathbf{v}_{E}=\mathbf{E}\times\mathbf{B}/B^{2}$ is the
$\mathbf{E}\times\mathbf{B}$ drift, $\phi$ is the potential, $J_{\parallel}$
is the parallel current, $\kappa=\mathbf{b}\cdot\nabla\mathbf{b}$ is the
magnetic curvature, $p$ is the pressure, $\nu$ is the collision frequency, $n$
is the total density and $\nabla_{\parallel}=\mathbf{b_{0}}\cdot\nabla$.

Since the adiabatic electron response dominates the zero-order potential and
the dipole potential appears in the next order, we solve the vorticity
equation for the azimuthally symmetric zero order
$(\phi_{0},m=0,k_{\parallel}=0)$ and first order dipole $(\phi_{1},m=1)$
component of the potential in the local cylindrical blob coordinates where
$\phi\propto\sum e^{-im\theta+ik_{\|}z}.$ Here, $\theta$ is the local
azimuthal angle ($\theta=0$ in the $e_{R}$ direction at the outboard midplane)
relative to the blob center and $z$ indicates the coordinate along the field
line.

In zero order, assuming adiabatic electrons, the right-hand side of Eq. (2)
and hence the parallel current vanishes. In the first order, the $\kappa$ term
in Eq. (1) acts as a source term for the m = 1 dipole potential. After some
algebra that combines the m = 1 first-order vorticity, density, and parallel
current equations, one obtains

$L_{1}\phi_{1}-k_{\parallel}^{2}\sigma_{\parallel,1}\phi_{1}=\frac{2c}{B}\mathbf{b}\times\kappa\cdot\nabla
p,$ (3)

where $L_{1}$ is an inertial term given by
$L_{1}=\frac{\omega_{pi}^{2}}{4\pi\Omega_{i}^{2}}\left(-i\Omega_{b}\nabla_{\perp}^{2}+\frac{c}{B}\nabla(\nabla^{2}_{\perp}\phi_{0})\cdot\mathbf{b}\times\nabla\right),$
and $\sigma_{\parallel,1}$ is the conductivity relating $J_{\parallel,1}$ to
$E_{\parallel,1}$

$\sigma_{\parallel,1}=i\frac{ne^{2}}{m_{e}}\frac{1-\alpha_{ad}}{\Omega_{b}+i\nu-
k_{\parallel}^{2}v_{te}^{2}/\Omega_{b}},$ (4)

Here $\Omega_{b}=\frac{c}{rB}\frac{d\phi_{0}}{dr}$ is the spin frequency,
$v_{te}$ is the thermal velocity and
$\alpha_{ad}=\frac{T_{e}}{en_{0}}\frac{dn_{0}}{d\phi_{0}}$ indicates the
adiabaticity. The $\alpha_{ad}$ parameter is introduced for the theoretical
description; the simulations employ the full drift-kinetic electron response.

Based on the above equations, we can solve for the propagation of spinning
blobs driven by the curvature mechanism which is well known in Eqs. 1 and 2 to
charge-polarize the blob. In the present formalism, when
$\sigma_{\parallel,1}$ is real (for example in the collisional limit)
$\mathrm{Re}[\phi_{1}\mathrm{exp}(-im\theta)]\propto\mathrm{cos}\theta$
results in an electric field in the $e_{Z}$ direction (up-down in the $RZ$
plane) and hence a radially outward $\mathbf{E}\times\mathbf{B}$ blob
propagation velocity. Poloidal blob propagation results from Eqs. 3 and 4 when
the left-hand side of Eq. 3 gives $\phi_{1}$ an imaginary part. In addition to
dipole charge polarization, the blob motion is also affected by the background
$\mathbf{E}\times\mathbf{B}$ flow, curvature, and grad-$B$ drifts.

Effect of the zonal field. To include the effect of the zonal field and the
resulting sheared flows, $\phi$ contains zero order terms describing the
background zonal flow $\phi_{0f}$ as well as the blob spin $\phi_{0}$. The
calculation is performed in the frame of the flow at the blob center so that
$\partial/\partial t$ can be neglected. The resulting equation for the first
order response $\phi_{1}+\phi_{1f}$ from the curvature and flow terms
respectively is then

$\begin{split}&(L_{1}-\sigma_{\parallel,1}k_{\parallel}^{2})(\phi_{1}+\phi_{1f})=\\\
&\frac{2}{c}\mathbf{b}\times\mathbf{\kappa}\cdot
p-\frac{\omega_{pi}^{2}}{4\pi\Omega_{i}^{2}}(\mathbf{v}_{E0}\cdot\nabla\nabla_{\perp}^{2}\phi_{0f}+\mathbf{v}_{E0f}\cdot\nabla\nabla^{2}_{\perp}\phi_{0})\\\
&\sim\frac{ne}{\omega_{i}\delta_{b}}\left[\frac{\delta
n}{n}\frac{2c_{s}^{2}}{R}+\frac{e\phi_{0}}{T_{e}}\left(\frac{\rho_{s}^{2}}{L_{f}^{2}}+\frac{\rho_{s}^{2}}{\delta_{b}^{2}}\right)\frac{eE_{0f}}{m_{i}}\right]\end{split}$
(5)

The first term on the right-hand side of Eq. 5 is the source of the charge
separation process, the second term is the source of the zonal field effect.
Here local Cartesian coordinates at the outboard midplane are employed where
$x$ is radial and $y$ is binormal,
$\mathbf{v}_{E0f}=v_{E0f}\mathbf{e}_{y}=(c/B)\mathbf{b}\times\nabla\phi_{0f}$,
and a small flow ordering $\nabla v_{E0f}\ll\Omega_{b}$ has been assumed to
neglect quadratic products of flow terms and to simplify the operators on the
left-hand side. Near the blob center at $x=0$, $v_{E0f}\approx x\partial
v_{E0f}/\partial x\sim\textrm{min}(\delta_{b},L_{f})v_{E0f}/L_{f}$, where
$\delta_{b}$ is the blob radius, assumed to be circular, and $L_{f}$ is the
scale length of the flow shear.

The size estimate of the terms on the right-hand side of Eq. 5 expresses the
sources in terms of the charge polarizing force per unit mass acting on the
spinning blob. As is well known[2] the curvature and grad-$B$ effects give
rise to the effective gravitational acceleration $g\sim 2c_{s}^{2}/R$. The
factor $\delta n/n$ takes into account the blob amplitude over the background
density[3].

The flow term in square brackets has two parts. The second part can be
expressed as
$e\phi_{0}\rho_{s}^{2}eE_{0f}/(T_{e}\delta^{2}_{b}m_{i})\sim(\Omega_{b}/\Omega_{i})(eE_{0f}/m_{i})$
which shows that the force on the blob is proportional to the blob vorticity
$\bar{\omega_{b}}\equiv\Omega_{b}$ times the zonal electric field force
$eE_{0f}/m_{i}$, i.e., the blob reacts to the zonal field like a (positive)
charge since charge is proportional to vorticity through the Poisson equation.
The actual interaction is complicated by the fact that in the flow frame,
$v_{E0f}$ and $E_{0f}$ change signs across the blob center, so that the blob
actually reacts to the velocity shear, as will be made explicit in Eq. 6.

The first term in square brackets can be rewritten as
$(\bar{\omega}_{f}/\Omega_{i})(e\phi_{0}/m_{i}\delta_{b})(\delta_{b}/L_{f})$
which shows the interaction of the blob electric field with the vorticity of
the flow, $\bar{\omega}=v_{E0f}/L_{f}$. The extra factor of $\delta_{b}/L_{f}$
makes the two terms symmetric noting that one factor of $\delta_{b}$ has been
pulled outside the square brackets.

Both of these terms show explicitly the mutual interaction of two different
sources of vorticity arising from the blob and the shear layer. Blob and shear
layer interactions have been studied previously in slab geometry fluid
simulations [24, 21] and experiments[25, 26]; however, the vortex analysis and
interpretation turn out to be particularly transparent in the present large
spin ordering.

The radial dependence of the zonal field, and the resulting shearing effect on
blobs, compete with spin which can stabilize the blob and preserve its
coherency. The stabilization criteria for spinning blobs from this competition
is $\Omega_{b}>\nabla v_{E0f}\sim v_{E0f}/L_{f}$. The shearing rate can also
affect the blob shape when it is large enough to distort the blobs as
indicated in the simulations. For instance, at $37.36\mathrm{\mu s}$ in
simulations, $\Omega_{b}\approx 1.70\times 10^{6}\mathrm{s^{-1}}$ is less than
$\nabla v_{E0f}\approx 1.95\times 10^{6}\mathrm{s^{-1}}$, resulting in the
blob bifurcation (Fig. 1). Thus, rapidly spinning blobs usually have a more
stable shape. Blob stability is widely observed in experiments[12, 13],
although direct experimental detection of blob spin has so far not been
possible.

Figure 3: The time evolution of zonal potential (left) and trajectories of the
center-of-mass blob location (right). In the left figure, the solid lines are
the zonal potential at different time and the black dashed line indicates the
separatrix. In the right figure, the solid line illustrates the time history
of the center-of-mass location in the $RZ$ plane; the colored dots indicate
the center-of-mass locations at different times; the dashed line indicates the
separatrix surface. Note that the zonal potential is due to particle loss in
the divertor. Figure 4: Scaling of the transport barrier criteria
$C_{\mathrm{TB}}$ versus radial velocity $v_{b}^{R}$ and averaged $\mathbf{\rm
E}\times\mathbf{\rm B}$ velocity $<v_{E}^{R}>$. In the scaling, we restart the
simulations at a selected time point $t=9.34\mu s$ with manually altered zonal
fields $[0.5,1,1.5,2,4,8]\phi_{f}$, corresponding to different
$C_{\mathrm{TB}}$. For $\phi_{f}$, $C_{\mathrm{TB}}\approx 0.727$ with
$L_{f}\approx 19\mathrm{mm}$, $\delta_{b}\approx 13\mathrm{mm}$,
$v^{\prime}_{E0f}\approx 4.40\times 10^{5}\mathrm{s^{-1}}$, $\Omega_{i}\approx
3.47\times 10^{7}\mathrm{rad/s}$ and $R\approx 0.890\mathrm{m}$. The cross
point of the vertical and horizontal dashed lines indicates the exact turning
point $C_{\mathrm{TB}}\approx 1.753$ in the simulations.

Blob transport barrier. The radial drift velocity induced by dipole response
due to the effect of the curvature drift charge separation is always outward,
while the radial drift velocity induced by the zonal field effect depends on
$E_{0f}$ and its shear, which can be negative or positive (Fig. 3). Thus, when
these two mechanisms compete to determine the radial velocity, a transport
barrier for the spinning blobs can be induced. By comparing the flow terms on
the right-hand side of Eq. 5 to the curvature term, it is possible to derive
the condition for a blob transport barrier. Recall that when the left-hand
side is real (dissipative) the curvature term drives the blob radially
outward, across the separatrix and into the SOL. The condition for the flow
terms to dominate the curvature term is

$C_{\mathrm{TB}}=\left(\frac{1}{L_{f}}+\frac{1}{\delta_{b}}\right)\frac{Rv^{\prime}_{E0f}}{2\Omega_{i}}>1,$
(6)

where the estimate $v_{E0f}=\mathrm{min}(L_{f},\delta_{b})v^{\prime}_{E0f}$
has been used to cancel one factor of $L_{f}$ or $\delta_{b}$ in the final
expression. See the comment following in Eq. 5. In deriving Eq. 6 the
condition $e\phi_{0}/T_{e}\sim\delta n/n$ for an adiabatic blob has been
employed. As shown in Fig. 4, for the simulation reported here, the inequality
in Eq. 6 is satisfied within factors of order unity, and the dynamics observed
in the simulation agrees well with our theoretical expectation for net radial
repulsion by the shear layer and shear-driven distortion and bifurcation of
the blob. Furthermore, as discussed subsequently, the transport barrier
condition is also relevant for experiments.

Comparison with experiments. Comparison between theory, simulations, and
experiments is a good tool to verify and validate the theory. We note the
following qualitative and quantitative correspondences. (a) The new
theoretical framework does not conflict with the dipole potential structure
observed in an experimental cross-correction analysis[27], since the dipole
potential is also included in the theory. (b) Blob bifurcation in the
separatrix region has been clearly observed in C-Mod[28, 29] and NSTX[30],
which agrees with our simulations and theoretical expectation. (c) Negative
radial motion of blobs in TCV [14] and NSTX [12] is potentially due to their
interaction with shear layers. (d) The zonal field effect in the edge region
is larger in H mode than in L mode. Thus blob emission into the SOL is
expected and measured to be lower in H mode than in L mode, for example, in
DIII-D [15] and NSTX[13]. A 3kHz oscillation in the edge zonal field region of
NSTX was found to be anti-correlated [31, 32] with blob emission into the SOL.
The above two experimental results are consistent with our theoretical
prediction. (e) According to the experimental data[28, 33] in C-Mod,
$L_{f}\approx 5\mathrm{mm}$, $\delta_{b}\approx 5\mathrm{mm}$,
$v^{\prime}_{E0f}\approx 2\times 10^{6}\mathrm{s^{-1}}$ for H mode, $5\times
10^{5}\mathrm{s^{-1}}$ for L mode, $\Omega_{i}\approx 2.25\times
10^{8}\mathrm{rad/s}$ and $R\approx 0.89\textrm{m}$ in the separatrix region,
$C_{\mathrm{TB}}\approx 1.09$ for H mode and $C_{\mathrm{TB}}\approx 0.27$ for
L mode, which shows a good quantitative agreement. (f) Recently a detailed
study of blob-vortex interactions with shear layers in L and H mode has been
carried out for the RFX-Mod device. The results[26] are in qualitative
agreement with the present theory. Thus, the present theoretical framework for
blob structure and motion is consistent with the experimentally observed
behavior of blobs.

Summary. In this letter, we investigated blob dynamics by seeding a blob as an
initial condition in the comprehensive full-f XGC particle code with
gyrokinetic ions, drift-kinetic electrons and realistic X-point geometry in
the electrostatic limit. We find that spin dominates the blob motion due to
the adiabatic electron response. The dipole potential induced by the non-
adiabatic part is smaller compared to the non-spinning case but is still
responsible for radially outward motion. We take advantage of this ordering to
develop a theory where the adiabatic potential is zeroth order and the dipole
potential is first order. This theory provides a good explanation of the
simulation results and it is able to explain several experimental
observations. A transport barrier criteria is obtained from the current theory
and the transport barrier condition in the simulation and from experimental
data is consistent with the behavior of blobs in the simulation and in
experiments. The current theoretical framework can be extended and enriched to
include kinetic and other effects, enabling more accurate descriptions of the
experimental behavior of blobs in magnetically confined devices.

It is noted that this work does not aim to model the buildup of the whole
pedestal. This work aims to isolate the spinning blobs and zonal field, and
investigate the zonal field effect on the spinning blobs. Although the
assumptions such as the blob seeding and flat profiles are used in our
simulations, these gyrokinetic simulations with realistic geometry and
equilibrium are first principles with the drift kinetic electrons, including
both adiabatic and non-adiabatic electron responses. In the future, we will
investigate the turbulent interaction of many blobs with realistic magnetic
geometry and profiles in different parameter regimes. We will also consider
different birth rates in L-mode and H-mode, and their impact on the SOL width,
leading to a more complete theoretical transport model.

Acknowledgements This research was supported by the Exascale Computing Project
(17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office
of Science and the National Nuclear Security Administration, and the SciDAC-4
project High-fidelity Boundary Plasma Simulation funded by the U.S. Department
of Energy Office of Science under Contract No. DE-AC02-09CH11466. This
research used resources from the Oak Ridge Leadership Computing Facility at
the Oak Ridge National Laboratory (ORNL) and resources of the National Energy
Research Scientific Computing Center (NERSC), which is supported by the Office
of Science of the U.S. Department of Energy under Contract Nos.
DEAC05–00OR22725 and DE-AC02–05CH11231, respectively.

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