# Turbulent field fluctuations in gyrokinetic and fluid plasmas

A. Mathews<EMAIL_ADDRESS>MIT Plasma Science and Fusion Center, Cambridge,
Massachusetts 02139, USA  N. Mandell  MIT Plasma Science and Fusion Center,
Cambridge, Massachusetts 02139, USA  M. Francisquez  Princeton Plasma Physics
Laboratory, Princeton, New Jersey 08540, USA  J.W. Hughes  MIT Plasma Science
and Fusion Center, Cambridge, Massachusetts 02139, USA  A. Hakim  Princeton
Plasma Physics Laboratory, Princeton, New Jersey 08540, USA

###### Abstract

A key uncertainty in the design and development of magnetic confinement fusion
energy reactors is predicting edge plasma turbulence. An essential step in
overcoming this uncertainty is the validation in accuracy of reduced turbulent
transport models. Drift-reduced Braginskii two-fluid theory is one such set of
reduced equations that has for decades simulated boundary plasmas in
experiment, but significant questions exist regarding its predictive ability.
To this end, using a novel physics-informed deep learning framework, we
demonstrate the first ever direct quantitative comparisons of turbulent field
fluctuations between electrostatic two-fluid theory and electromagnetic
gyrokinetic modelling with good overall agreement found in magnetized helical
plasmas at low normalized pressure. This framework presents a new technique
for the numerical validation and discovery of reduced global plasma turbulence
models.

††preprint: AIP/123-QED

##  I Introduction

A hallmark of a turbulence theory is the nonlinear connection that it sets
between dynamical variables. For ionized gases, perturbations in the plasma
are intrinsically linked with perturbations of the electromagnetic field.
Accordingly, the electric potential response concomitant with electron
pressure fluctuations constitutes one of the key relationships demarcating a
plasma turbulence theory and presents a significant consistency test en route
to its validation. But to meaningfully evaluate this dynamical interplay
across theories or experiment using standard numerical methods is an intricate
task. Namely, when comparing global plasma simulations to other numerical
models or diagnostics, one must precisely align initial states, boundary
conditions, and sources of particles, momentum, and energy (e.g. plasma
heating, sheath effects, atomic and molecular processes) Francisquez  _et al._
(2020a); Mijin  _et al._ (2020). These descriptions of physics can be
increasingly difficult to reconcile when applying different
representations—for example, fluid or gyrokinetic—or when measurements
characterizing the plasma are sparse as commonly encountered in thermonuclear
fusion devices. Additionally, imperfect conservation properties (both in
theory and numerics due to discretization) and the limited accuracy of time-
integration schema can introduce errors further misaligning comparative
modelling efforts. Such difficulties exist even when examining the same
nominal turbulence theory across numerical codes Görler  _et al._ (2016);
Tronko  _et al._ (2017); Merlo  _et al._ (2018); Mikkelsen  _et al._ (2018)
and become magnified for cross-model comparisons Francisquez  _et al._
(2020a). To overcome these classical limitations, we apply a custom physics-
informed deep learning framework Mathews  _et al._ (2021a) with a continuous
spatiotemporal domain and partial electron pressure observations to
demonstrate the first direct comparisons of instantaneous turbulent fields
between two distinct global plasma turbulence models: electrostatic drift-
reduced Braginskii theory and electromagnetic long-wavelength gyrokinetics.

This is significant since validating the accuracy of reduced turbulence
theories is amongst the greatest challenges to the advancement of plasma
modelling. In the fusion community, both comprehensive full-$f$ global
gyrokinetic Mandell  _et al._ (2020); Hager  _et al._ (2017); Pan  _et al._
(2018); Dorf  _et al._ (2016); Chôné  _et al._ (2018); Lanti  _et al._ (2020);
Grandgirard  _et al._ (2016); Boesl  _et al._ (2019) and two-fluid Dudson  _et
al._ (2009); Tamain  _et al._ (2016); Halpern  _et al._ (2016); Stegmeir  _et
al._ (2018); Zhu, Francisquez, and Rogers (2018) direct numerical simulations
are under active development to improve predictive modelling capabilities and
the design of future reactors, but no single code can currently capture all
the dynamics present in the edge with sufficient numerical realism. It is
therefore of paramount importance to recognize when certain numerical or
analytical simplifications make no discernible difference, or when a given
theoretical assumption is too risky—to clearly identify the limits of present
simulations Francisquez  _et al._ (2020a). Validating reduced plasma
turbulence models is accordingly essential. Adaptations of the drift-reduced
Braginskii equations have recently investigated several important edge
phenomena Francisquez, Zhu, and Rogers (2017); Ricci  _et al._ (2012); Thrysøe
_et al._ (2018); Nespoli  _et al._ (2017), but precise quantitative agreement
with experiment is generally lacking on a wide scale and their accuracy in
fusion reactors is uncertain, especially since kinetic effects may not be
negligible as edge plasma dynamics approach the ion Larmor radius and
collision frequencies are comparable to turbulent timescales Braginskii
(1965); D’Ippolito, Myra, and Zweben (2011). Yet these fluid codes remain in
widespread usage for their reduced computational cost, which offers the
ability to perform parameter scans with direct numerical simulations that can
help uncover underlying physics and optimize reactor engineering designs.
Nevertheless, the Vlasov-Maxwell system of equations is the gold standard in
plasma theory, but such modelling is prohibitively expensive when wholly
simulating the extreme spatiotemporal scales extant in fusion reactors. For
the purposes of more tractable yet high fidelity numerical simulations, this
system is averaged over the fast gyro-motion of particles to formulate 5D
gyrokinetics. Fluid-gyrokinetic comparisons therefore present a way to analyze
the robustness of reduced modelling while also illuminating when collisional
fluid theories may be insufficient. Quantifying potential shortcomings is thus
vital for the testing and development of improved reduced models Waltz  _et
al._ (2019).

A particularly important quantity for testing predictive modelling is edge
electric field fluctuations since turbulent ${\bf E\times B}$ drifts strongly
influence edge plasma stability and transport of blob-like structures which
can account for over 50% of particle losses during standard operation of
tokamaks D’Ippolito, Myra, and Zweben (2011); Simakov and Catto (2003); Wang
_et al._ (2020); Zhang, Guo, and Diamond (2020); Carralero  _et al._ (2017);
Kube  _et al._ (2018). Resultant turbulent fluxes impacting the walls can
cause sputtering, erosion, and impurity injection which adversely affect safe
operation and confinement of fusion plasmas D’Ippolito, Myra, and Zweben
(2011); Kuang  _et al._ (2020, 2018). While explicit comparisons of global
nonlinear dynamics across turbulence models were previously impractical using
standard numerical simulations, our physics-informed deep learning Raissi,
Perdikaris, and Karniadakis (2019) technique enables direct quantitative
cross-examination of plasma turbulence theories by uncovering the turbulent
electric field expected in two-fluid and gyrokinetic models based upon an
electron pressure fluctuation Mathews  _et al._ (2021a). This validation
framework is potentially transferable to magnetized plasma turbulence in
contexts beyond fusion energy such as space propulsion systems and
astronomical settings as well. Overall, these physics-informed neural networks
present a new paradigm in the advancement of reduced plasma turbulence models.

To demonstrate these results, we proceed with a description of gyrokinetic
modelling based upon Mandell  _et al._ (2020) in Section II, outline a custom
physics-informed machine learning architecture from Mathews  _et al._ (2021a)
suited for the analysis of two-fluid theory in Section III, present the first
direct comparisons of instantaneous turbulent fields in section IV, and
conclude with a summary and future outlook in Section V.

##  II  Gyrokinetic modelling

Following the full treatment outlined in Mandell (2021) and Mandell  _et al._
(2020), the long-wavelength gyrokinetic modelling analyzed constitutes
nonlinear global electromagnetic turbulence simulations on open field lines
using the Gkeyll computational plasma framework. The full-$f$ electromagnetic
gyrokinetic equation is solved in the symplectic formulation Brizard and Hahm
(2007), which describes the evolution of the gyrocenter distribution function
$f_{s}(\mathbf{Z},t)=f_{s}(\mathbf{R},v_{\parallel},\mu,t)$ for each species
($s=\\{e,i\\}$), where $\mathbf{Z}$ is a phase-space coordinate composed of
the guiding center position $\mathbf{R}=(x,y,z)$, parallel velocity
$v_{\parallel}$, and magnetic moment $\mu=m_{s}v_{\perp}^{2}/2B$. In terms of
the gyrocentre Hamiltonian and the Poisson bracket in gyrocentre coordinates
with collisions $C[f_{s}]$ and sources $S_{s}$, the gyrokinetic equation is

$\displaystyle\frac{\partial f_{s}}{\partial
t}+\\{f_{s},H_{s}\\}-\frac{q_{s}}{m_{s}}\frac{\partial A_{\parallel}}{\partial
t}\frac{\partial f_{s}}{\partial v_{\parallel}}=C[f_{s}]+S_{s},$ (1)

or, equivalently,

$\displaystyle\frac{\partial f_{s}}{\partial
t}+\dot{\mathbf{R}}\mathbf{\cdot}\nabla
f_{s}+\dot{v}^{H}_{\parallel}\frac{\partial f_{s}}{\partial
v_{\parallel}}-\frac{q_{s}}{m_{s}}\frac{\partial A_{\parallel}}{\partial
t}\frac{\partial f_{s}}{\partial v_{\parallel}}=C[f_{s}]+S_{s},$ (2)

where the gyrokinetic Poisson bracket is defined as

$\\{F,G\\}=\frac{\mathbf{B^{*}}}{m_{s}B_{\parallel}^{*}}\mathbf{\cdot}\left(\nabla
F\frac{\partial G}{\partial v_{\parallel}}-\frac{\partial F}{\partial
v_{\parallel}}\nabla
G\right)-\frac{\mathbf{\hat{b}}}{q_{s}B_{\parallel}^{*}}\times\nabla
F\mathbf{\cdot}\nabla G,$ (3)

and the gyrocentre Hamiltonian is

$H_{s}=\frac{1}{2}m_{s}v_{\parallel}^{2}+\mu B+q_{s}\phi.$ (4)

The nonlinear phase-space characteristics are given by

$\dot{\mathbf{R}}=\\{\mathbf{R},H_{s}\\}=\frac{\mathbf{B^{*}}}{B_{\parallel}^{*}}v_{\parallel}+\frac{\mathbf{\hat{b}}}{q_{s}B_{\parallel}^{*}}\times\left(\mu\nabla
B+q_{s}\nabla\phi\right),$ (5) $\displaystyle\dot{v}_{\parallel}$
$\displaystyle=\dot{v}^{H}_{\parallel}-\frac{q_{s}}{m_{s}}\frac{\partial
A_{\parallel}}{\partial
t}=\\{v_{\parallel},H_{s}\\}-\frac{q_{s}}{m_{s}}\frac{\partial
A_{\parallel}}{\partial t}$ (6)
$\displaystyle=-\frac{\mathbf{B^{*}}}{m_{s}B_{\parallel}^{*}}\mathbf{\cdot}\left(\mu\nabla
B+q_{s}\nabla\phi\right)-\frac{q_{s}}{m_{s}}\frac{\partial
A_{\parallel}}{\partial t}.$

Here, $B_{\parallel}^{*}=\mathbf{\hat{b}}\mathbf{\cdot}\mathbf{B^{*}}$ is the
parallel component of the effective magnetic field
$\mathbf{B^{*}}=\mathbf{B}+(m_{s}v_{\parallel}/q_{s})\nabla\times\mathbf{\hat{b}}+\delta\mathbf{B}$,
where $\mathbf{B}=B\mathbf{\hat{b}}$ is the equilibrium magnetic field and
$\delta\mathbf{B}=\nabla\times(A_{\parallel}\mathbf{\hat{b}})\approx\nabla
A_{\parallel}\times\mathbf{\hat{b}}$ is the perturbed magnetic field (assuming
that the equilibrium magnetic field varies on spatial scales larger than
perturbations so that $A_{\parallel}\nabla\times\mathbf{\hat{b}}$ can be
neglected). Higher-order parallel compressional fluctuations of the magnetic
field are neglected so that $\delta\mathbf{B}=\delta\mathbf{B}_{\perp}$ and
the electric field is given by
${\mathbf{E}}=-\nabla\phi-(\partial{A_{\parallel}}/\partial
t)\mathbf{\hat{b}}$. The species charge and mass are $q_{s}$ and $m_{s}$,
respectively. In (6), $\dot{v}_{\parallel}$ has been separated into a term
coming from the Hamiltonian,
$\dot{v}^{H}_{\parallel}=\\{v_{\parallel},H_{s}\\}$, and another term
proportional to the inductive component of the parallel electric field,
$(q_{s}/m_{s})\partial{A_{\parallel}}/\partial{t}$. In the absence of
collisions and sources, (1) can be identified as a Liouville equation
demonstrating that the distribution function is conserved along the nonlinear
phase space characteristics. Accordingly, the gyrokinetic equation can be
recast in the following conservative form,

$\displaystyle\frac{\partial(\mathcal{J}f_{s})}{\partial t}$
$\displaystyle+\nabla\mathbf{\cdot}(\mathcal{J}\dot{\mathbf{R}}f_{s})+\frac{\partial}{\partial
v_{\parallel}}\left(\mathcal{J}\dot{v}^{H}_{\parallel}f_{s}\right)$ (7)
$\displaystyle-\frac{\partial}{\partial
v_{\parallel}}\left(\mathcal{J}\frac{q_{s}}{m_{s}}\frac{\partial
A_{\parallel}}{\partial t}f_{s}\right)=\mathcal{J}C[f_{s}]+\mathcal{J}S_{s},$

where $\mathcal{J}=B_{\parallel}^{*}$ is the Jacobian of the gyrocentre
coordinates and
$\mathbf{\hat{b}}\mathbf{\cdot}\nabla\times\mathbf{\hat{b}}\approx 0$ so that
$B_{\parallel}^{*}\approx B$. The symplectic formulation of electromagnetic
gyrokinetics is utilized with the parallel velocity as an independent variable
instead of the parallel canonical momentum $p_{\parallel}$ in the Hamiltonian
formulation Brizard and Hahm (2007); Hahm, Lee, and Brizard (1988). We use
this notation for convenience, and to explicitly display the time derivative
of $A_{\parallel}$, which is characteristic of the symplectic formulation of
electromagnetic gyrokinetics. The electrostatic potential, $\phi$, is
determined by quasi-neutrality,

$\sigma_{g}+\sigma_{\text{pol}}=\sigma_{g}-\nabla\mathbf{\cdot}\mathbf{P}=0,$
(8)

with the guiding centre charge density (neglecting gyroaveraging in the long-
wavelength limit) given by

$\sigma_{g}=\sum_{s}q_{s}\int d\mathbf{w}\ \mathcal{J}f_{s}.$ (9)

Here,
$d\mathbf{w}=2\pi\,m_{s}^{-1}dv_{\parallel}\,d\mu=m_{s}^{-1}dv_{\parallel}\,d\mu\int
d\alpha$ is the gyrocentre velocity-space volume element $(d{\bf
v}=m_{s}^{-1}dv_{\parallel}\,d\mu\,d\alpha\,\mathcal{J})$ with the gyroangle
$\alpha$ integrated away and the Jacobian factored out (formally,
$\mathcal{J}$ should also be included in $d\mathbf{w}$). The polarization
vector is then

$\displaystyle\mathbf{P}$ $\displaystyle=-\sum_{s}\int d\mathbf{w}\
\frac{m_{s}}{B^{2}}\mathcal{J}f_{s}\nabla_{\perp}\phi$ (10)
$\displaystyle\approx-\sum_{s}\frac{m_{s}n_{0s}}{B^{2}}\nabla_{\perp}\phi\equiv-\epsilon_{\perp}\nabla_{\perp}\phi,$

where
$\nabla_{\perp}=\nabla-\mathbf{\hat{b}}(\mathbf{\hat{b}}\mathbf{\cdot}\nabla)$
is the gradient orthogonal to the background magnetic field. A linearized
time-independent polarization density, $n_{0}$, is assumed which is consistent
with neglecting a second-order ${\bf E\times B}$ energy term in the
Hamiltonian. Such an approximation in the scrape-off layer (SOL) is
questionable due to the presence of large density fluctuations, although a
linearized polarization density is commonly used in full-$f$ gyrokinetic
simulations for computational efficiency and reflective of common numerical
modelling practices Ku  _et al._ (2018); Shi  _et al._ (2019); Mandell  _et
al._ (2020). Adding the nonlinear polarization density along with the second-
order ${\bf E\times B}$ energy term in the Hamiltonian are improvements kept
for future work. Consequently, the quasi-neutrality condition can be rewritten
as the long-wavelength gyrokinetic Poisson equation,

$-\nabla\mathbf{\cdot}\sum_{s}\frac{m_{s}n_{0s}}{B^{2}}\nabla_{\perp}\phi=\sum_{s}q_{s}\int
d\mathbf{w}\ \mathcal{J}f_{s},$ (11)

where, even in the long-wavelength limit with no gyroaveraging, the first-
order polarization charge density on the left-hand side of (11) incorporates
some finite Larmor radius (FLR) or $k_{\perp}$ effects in its calculation. It
is worth emphasizing that this “long-wavelength” limit is a valid limit of the
full-$f$ gyrokinetic derivation since care was taken to include the guiding-
center components of the field perturbations at $O(1)$. Further, although one
may think of this as a drift-kinetic limit, the presence of the linearized ion
polarization term in the quasineutrality equation distinguishes the long-
wavelength gyrokinetic model from versions of drift-kinetics that, for
example, include the polarization drift in the equations of motion.

The parallel magnetic vector potential, $A_{\parallel}$, is determined by the
parallel Ampère equation,

$-\nabla_{\perp}^{2}A_{\parallel}=\mu_{0}\sum_{s}q_{s}m_{s}\int
v_{\parallel}\mathcal{J}f_{s}\,d\mathbf{w}.$ (12)

Note that we can also take the time derivative of this equation to get a
generalized Ohm’s law which can be solved directly for $\partial
A_{\parallel}/\partial t$, the inductive component of the parallel electric
field $E_{\parallel}$ Reynders (1993); Cummings (1994); Chen and Parker
(2001):

$-\nabla_{\perp}^{2}\frac{\partial A_{\parallel}}{\partial
t}=\mu_{0}\sum_{s}q_{s}m_{s}\int
v_{\parallel}\frac{\partial(\mathcal{J}f_{s})}{\partial t}\,d\mathbf{w}.$ (13)

Writing the gyrokinetic equation as

$\frac{\partial(\mathcal{J}f_{s})}{\partial
t}=\frac{\partial(\mathcal{J}f_{s})}{\partial
t}^{\star}+\frac{\partial}{\partial
v_{\parallel}}\left(\mathcal{J}\frac{q_{s}}{m_{s}}\frac{\partial
A_{\parallel}}{\partial t}f_{s}\right),$ (14)

where $\partial{(\mathcal{J}f_{s})^{\star}}/\partial{t}$ denotes all terms in
the gyrokinetic equation (including sources and collisions) except $\partial
A_{\parallel}/\partial t$ term, Ohm’s law can be rewritten after an
integration by parts as

$\displaystyle\left(-\nabla_{\perp}^{2}+\sum_{s}\mu_{0}q_{s}^{2}\int\mathcal{J}f_{s}\,d\mathbf{w}\right)\frac{\partial
A_{\parallel}}{\partial t}$
$\displaystyle\qquad\qquad=\mu_{0}\sum_{s}q_{s}m_{s}\int
v_{\parallel}\frac{\partial(\mathcal{J}f_{s})}{\partial
t}^{\star}\,d\mathbf{w}.$ (15)

To model collisions, we use a conservative Lenard–Bernstein (or Dougherty)
operator Lenard and Bernstein (1958); Dougherty (1964); Francisquez  _et al._
(2020b),

$\displaystyle\mathcal{J}C[f_{s}]$
$\displaystyle=\nu\left\\{\frac{\partial}{\partial
v_{\parallel}}\left[\left(v_{\parallel}-u_{\parallel}\right)\mathcal{J}f_{s}+v_{th,s}^{2}\frac{\partial(\mathcal{J}f_{s})}{\partial
v_{\parallel}}\right]\right.$ (16)
$\displaystyle\left.\quad+\frac{\partial}{\partial\mu}\left[2\mu\mathcal{J}f_{s}+2\mu\frac{m_{s}}{B}v_{th,s}^{2}\frac{\partial(\mathcal{J}f_{s})}{\partial\mu}\right]\right\\},$

where $n_{s}u_{\parallel}^{2}=\int
d\mathbf{w}\mathcal{J}v_{\parallel}^{2}f_{s}$, $3n_{s}v_{th,s}^{2}=2\int
d\mathbf{w}\mathcal{J}\mu Bf_{s}/m_{s}$, $n_{s}u_{\parallel}=\int
d\mathbf{w}\mathcal{J}v_{\parallel}f_{s}$, $n_{s}=\int
d\mathbf{w}\mathcal{J}f_{s}$, and $T_{s}=m_{s}v_{th,s}^{2}$. This collision
operator contains the effects of drag and pitch-angle scattering, and it
conserves number, momentum, and energy density. Consistent with our present
long-wavelength treatment of the gyrokinetic system, FLR effects are ignored.
In this work, both like-species and cross-species collisions among electrons
and ions are included. The collision frequency $\nu$ is kept velocity-
independent, i.e. $\nu\neq\nu(v)$. Further details about this collision
operator, including its conservation properties and discretization, can be
found in Francisquez  _et al._ (2020b).

To clarify the approximations undertaken in deriving the gyrokinetic model
formulated above and its consequent effects on turbulent fields, the key
assumptions are reviewed: The orderings in gyrokinetic theory that effectively
reduce the full phase space’s dimensionality are ${\omega}/{\Omega_{s}}\ll 1$
and $k_{\perp}/k_{\parallel}\gg 1$. These orderings express that the charged
particle gyrofrequency ($\Omega_{s}=q_{s}B/m_{s}$) in a magnetic field is far
greater than the characteristic frequencies of interest ($\omega$) with
perpendicular wavenumbers ($k_{\perp}$) of Fourier modes being far larger than
parallel wavenumbers ($k_{\parallel}$). Such properties are generally expected
and observed for drift-wave turbulence in magnetically confined fusion plasmas
where $\omega\ll\Omega_{s}$ Zweben  _et al._ (2007). An additional “weak-flow”
ordering Dimits, LoDestro, and Dubin (1992); Parra and Catto (2008); Dimits
(2012) is applied where $v_{\bf E\times B}/v_{th,s}\approx
k_{\perp}\rho_{s}q_{s}\phi/T_{s}\ll 1$ which allows for large amplitude
perturbations. This approximation is also generally valid for electrons and
deuterium ions in edge tokamak plasmas Belli and Candy (2018) as it assumes
${\bf E\times B}$ flows are far smaller than the thermal velocity, $v_{th,s}$,
as observed in experiment D’Ippolito, Myra, and Zweben (2011); Zweben  _et
al._ (2015). By constraining gradients of $\phi$ instead of $\phi$ itself,
this weak-flow ordering simultaneously allows perturbations of order unity
($q_{s}\phi/T_{s}\sim 1$) at long wavelengths ($k_{\perp}\rho_{s}\ll 1$) and
small perturbations ($q_{s}\phi/T_{s}\ll 1$) at short wavelengths
($k_{\perp}\rho_{s}\sim 1$) along with perturbations at intermediate scales.
Alternatively, this approximation can be intuitively viewed as the potential
energy variation around a gyro-orbit being small compared to the kinetic
energy, $q_{s}\phi(\mathbf{R}+{\bm{\rho}_{s}})-q_{s}\phi(\mathbf{R})\approx
q_{s}{\bm{\rho}_{s}}\cdot\nabla_{\perp}\phi\ll T_{s}$ Mandell (2021). Here
${\bm{\rho}_{s}}$ is the gyroradius vector which points from the center of the
gyro-orbit $\mathbf{R}$ to the particle location $\mathbf{x}$. To ensure
consistency in the gyrokinetic model at higher order (although the guiding-
centre limit is eventually taken in the continuum simulations, i.e.
$k_{\perp}\rho_{s}\ll 1$), a strong-gradient ordering is also employed which
assumes the background magnetic field varies slowly relative to edge profiles
Zweben  _et al._ (2007). As noted above, we have taken the long-wavelength
limit and neglected variations of $\phi$ on the scale of the gyroradius. This
yields guiding-center equations of motion which do not contain gyroaveraging
operations. While extensions to a more complete gyrokinetic model are in
progress, these contemporary modelling limitations are worth noting for the
scope of the present results.

In accordance with Mandell  _et al._ (2020), the gyrokinetic turbulence is
simulated on helical, open field lines as a rough model of the tokamak scrape-
off layer at NSTX-like parameters. A non-orthogonal, field-aligned geometry
Beer, Cowley, and Hammett (1995) is employed for numerical modelling whereby
$x$ is the radial coordinate, $z$ is the coordinate parallel to the field
lines, and $y$ is the binormal coordinate which labels field lines at constant
$x$ and $z$. These coordinates map to physical cylindrical coordinates
($R,\varphi,Z)$ via $R=x$, $\varphi=(y/\sin\theta+z\cos\theta)/R_{c}$,
$Z=z\sin\theta$. The field-line pitch $\sin\theta=B_{v}/B$ is taken to be
constant, with $B_{v}$ the vertical component of the magnetic field (analogous
to the poloidal field in tokamaks), and $B$ the total magnitude of the
background magnetic field. The open field lines strike material walls at the
top and bottom of the domain consistent with the simple magnetized torus
configuration studied experimentally via devices such as the Helimak Gentle
and He (2008) and TORPEX Fasoli  _et al._ (2006). This system without magnetic
shear contains unfavorable magnetic curvature producing the interchange
instability that drives edge turbulence. There is no good curvature region to
produce conventional ballooning-mode structure in the current setup. Further,
$R_{c}=R_{0}+a$ is the radius of curvature at the centre of the simulation
domain, with $R_{0}$ the major radius and $a$ the minor radius. The curvature
operator is

$(\nabla\times\mathbf{\hat{b}})\mathbf{\cdot}\nabla
f(x,y,z)\approx\left[(\nabla\times\mathbf{\hat{b}})\mathbf{\cdot}\nabla
y\right]\frac{\partial f}{\partial y}=-\frac{1}{x}\frac{\partial f}{\partial
y},$ (17)

where $\mathbf{B}=B_{\text{axis}}(R_{0}/R)\mathbf{\hat{e}}_{z}$ in the last
step and $B_{\text{axis}}$ is the magnetic field strength at the magnetic
axis. This geometry represents a flux-tube-like domain on the outboard
strictly bad curvature side that wraps helically around the torus and
terminates on conducting plates at each end in $z$. Note that although the
simulation is on a flux-tube-like domain, it is not performed in the local
limit commonly applied in $\delta f$ gyrokinetic codes; instead, the
simulations are effectively global as they include radial variation of the
magnetic field and kinetic profiles. The simulation box is centred at
$(x,y,z)=(R_{c},0,0)$ with dimensions $L_{x}=56\rho_{i0}\approx 16.6$ cm,
$L_{y}=100\rho_{i0}\approx 29.1$ cm, and $L_{z}=L_{p}/\sin\theta=8.0$ m, where
$\rho_{i0}=\sqrt{m_{i}T_{i0}}/q_{i}B_{0}$ and $L_{p}=2.4$ m approximates the
vertical height of the SOL. The velocity-space grid has extents
$-4v_{th,s0}\leq v_{\parallel}\leq 4v_{th,s0}$ and $0\leq\mu\leq
6T_{s0}/B_{0}$, where $v_{th,s0}=\sqrt{T_{s0}/m_{s}}$ and
$B_{0}=B_{\text{axis}}R_{0}/R_{c}$. The low-$\beta$ simulations presented here
use piecewise-linear ($p=1$) basis functions, with
$(N_{x},N_{y},N_{z},N_{v_{\parallel}},N_{\mu})=(32,64,16,10,5)$ the number of
cells in each dimension. At high-$\beta$, due to the increased computational
cost, the resolution is lowered to
$(N_{x},N_{y},N_{z},N_{v_{\parallel}},N_{\mu})=(16,32,14,10,5)$. For $p=1$,
one should double these numbers to obtain the equivalent number of grid points
for comparison with standard grid-based gyrokinetic codes.

The radial boundary conditions model conducting walls at the radial ends of
the domain, given by the Dirichlet boundary condition $\phi=A_{\parallel}=0$.
The condition $\phi=0$ effectively prevents ${\bf E\times B}$ flows into
walls, while $A_{\parallel}=0$ makes it so that (perturbed) field lines never
intersect the walls. For the latter, one can think of image currents in the
conducting wall that mirror currents in the domain, resulting in exact
cancellation of the perpendicular magnetic fluctuations at the wall. Also, the
magnetic curvature and $\nabla B$ drifts do not have a radial component in
this helical magnetic geometry. These boundary conditions on the fields are
thus sufficient to guarantee that there is no flux of the distribution
function into the radial walls. Conducting-sheath boundary conditions are
applied in the $z$-direction Shi  _et al._ (2017, 2019) to model the Debye
sheath (the dynamics of which is beyond the gyrokinetic ordering), which
partially reflects one species (typically electrons) and fully absorbs the
other species depending on the sign of the sheath potential. This involves
solving the gyrokinetic Poisson equation for $\phi$ at the $z$-boundary (i.e.
sheath entrance), and using the resulting sheath potential to determine a
cutoff velocity below which particles (typically low energy electrons) are
reflected by the sheath. Notably, this boundary condition allows current
fluctuations in and out of the sheath. This differs from the standard logical
sheath boundary condition Parker  _et al._ (1993) which imposes zero net
current to the sheath by assuming ion and electron currents at the sheath
entrance are equal at all times. The fields do not require a boundary
condition in the $z$-direction since only perpendicular derivatives appear in
the field equations. The simulations are carried out in a sheath-limited
regime but there can be electrical disconnection from the plasma sheath if the
Alfvén speed is slow enough. Periodic boundary conditions are used in the
$y$-direction. The simulation parameters roughly approximate an H-mode
deuterium plasma in the NSTX SOL: $B_{\text{axis}}=0.5$ T, $R_{0}=0.85$ m,
$a=0.5$ m, $T_{e0}=T_{i0}=40$ eV. To model particles and heat from the core
crossing the separatrix, we apply a non-drifting Maxwellian source of ions and
electrons,

$S_{s}=\frac{n_{S}(x,z)}{(2\pi
T_{S}/m_{s})^{3/2}}\exp\left(-\frac{m_{s}v^{2}}{2T_{S}}\right),$ (18)

with source temperature $T_{S}=70$ eV for both species and
$v^{2}=v_{\parallel}^{2}+2\mu B/m_{s}$. The source density is given by

$n_{S}(x,z)=\begin{cases}S_{0}\exp\left(\frac{-(x-x_{S})^{2}}{(2\lambda_{S})^{2}}\right)\qquad\qquad&|z|<L_{z}/4\\\
0\qquad&\mathrm{otherwise}\end{cases}$ (19)

so that $x_{S}-3\lambda_{S}<x<x_{S}+3\lambda_{S}$ delimits the source region,
and we take $x_{S}=1.3$ m and $\lambda_{S}=0.005$ m. This localized particle
source structure in $z$-space results in plasma ballooning out largely in the
centre of the magnetic field line. The source particle rate $S_{0}$ is chosen
so that the total (ion plus electron) source power matches the desired power
into the simulation domain, $P_{\mathrm{src}}$. Since we only simulate a flux-
tube-like fraction of the whole SOL domain, $P_{\mathrm{src}}$ is related to
the total SOL power, $P_{\mathrm{SOL}}$, by
$P_{\mathrm{src}}=P_{\mathrm{SOL}}L_{y}L_{z}/(2\pi
R_{c}L_{\mathrm{pol}})\approx 0.115P_{\mathrm{SOL}}$. Amplitudes are adjusted
to approximate $P_{\mathrm{SOL}}=5.4$ MW crossing into these open flux
surfaces at low-$\beta$ conditions relevant to NSTX Zweben  _et al._ (2015).
An artificially elevated density case with $P_{\mathrm{SOL}}=54.0$ MW is also
tested to study edge turbulence at high-$\beta$. The collision frequency is
comparable in magnitude to the inverse autocorrelation time of electron
density fluctuations at low-$\beta$. For the high-$\beta$ case, $\nu$ is found
to be about $10\times$ larger, and the plasma thus sits in a strongly
collisional regime. Simulations reach a quasi-steady state with the sources
balanced by end losses to the sheath, although there is no neutral recycling
included yet which is a focus of ongoing work Bernard (2020). Unlike in Shi
_et al._ (2019), no numerical heating nor source floors are applied in the
algorithm to ensure positivity.

In summary, the full-$f$ nonlinear electromagnetic gyrokinetic turbulence
simulations of the NSTX plasma boundary region employ the lowest-order, i.e.
guiding-center or drift-kinetic limit, of the system. Implementing
gyroaveraging effects given by the next order terms in advanced geometries is
the focus of future work. These present approximations in modern full-$f$
global gyrokinetic simulations should be kept in mind when attributing any
similarities or differences to two-fluid theory in the next sections since the
gyrokinetic formulation can itself be improved. The turbulence simulations
presented are thus a reflection of the current forefront of numerical
modelling. A full exposition of the derivation and benchmarking including the
energy-conserving discontinuous Galerkin scheme applied in Gkeyll for the
discretization of the gyrokinetic system in 5-dimensional phase space along
with explicit time-stepping and avoidance of the Ampère cancellation problem
can be found in Mandell (2021).

##  III Machine learning fluid theory

A fundamental goal in computational plasma physics is determining the minimal
complexity necessary in a theory to sufficiently represent (anomalous)
observations of interest for predictive fusion reactor simulations.
Convection-diffusion equations with effective mean transport coefficients are
widely utilized Reiter  _et al._ (1991); Dekeyser  _et al._ (2019) but
insufficient in capturing edge plasma turbulence where scale separation
between the equilibrium and fluctuations is not justified Naulin (2007);
Mandell (2021). Other reduced turbulence models such as classical
magnetohydrodynamics are unable to resolve electron and ion temperature
dynamics with differing energy transport mechanisms in the edge
Ko$\mathbf{c}$an  _et al._ (2008, 2009); Nielsen  _et al._ (2016). Following
the framework set forth in Mathews  _et al._ (2021a), we consider here the
widely used first-principles-based two-fluid drift-reduced Braginskii
equations Braginskii (1965); Francisquez (2018) in the electrostatic limit
relevant to low-$\beta$ conditions in the edge of fusion experiments Zweben
_et al._ (2015) for comparison to gyrokinetic modelling Mandell  _et al._
(2020). This is also a full-$f$ Belli and Candy (2008); Madsen (2013); Held,
Wiesenberger, and Kendl (2020) nonlinear model in the sense that the evolution
of the equilibrium and fluctuating components of the solution are merged and
relative perturbation amplitudes can be of order unity but evaluated in the
fluid limit. This two-fluid theory assumes the plasma is magnetized
($\Omega_{i}\gg\frac{\partial}{\partial t}$), collisional
($\nu_{ei}\gg\frac{\partial}{\partial t}$), and quasineutral
($\nabla\cdot\mathbf{j}\approx 0$) with the reduced perpendicular fluid
velocity given by ${\bf E\times B}$, diamagnetic, and ion polarization drifts.
After neglecting collisional drifts and terms of order $m_{e}/m_{i}$, one
arrives at the following set of equations (in Gaussian units) governing the
plasma’s density ($n\approx n_{e}\approx n_{i}$), vorticity ($\omega$),
parallel electron velocity ($v_{\parallel e}$), parallel ion velocity
($v_{\parallel i}$), electron temperature ($T_{e}$), and ion temperature
($T_{i}$) Francisquez  _et al._ (2020a); Francisquez (2018)

$\displaystyle\frac{d^{e}n}{dt}=-\frac{2c}{B}\left[n{C}_{\left({\phi}\right)}-\frac{1}{e}{C}_{\left({p_{e}}\right)}\right]-n\nabla_{\parallel}v_{\parallel
e}+S_{n}+\mathcal{D}_{n}$ (20) $\displaystyle\frac{\partial\omega}{\partial
t}$
$\displaystyle=\frac{2c}{eB}\left[{C}_{\left({p_{e}}\right)}+{C}_{\left({p_{i}}\right)}\right]-\frac{1}{em_{i}\Omega_{i}}{C}_{\left({G_{i}}\right)}$
(21)
$\displaystyle\quad+\frac{1}{e}\nabla_{\parallel}j_{\parallel}-\nabla\cdot\left\\{\frac{nc^{2}}{\Omega_{i}B^{2}}\left[\phi,\nabla_{\perp}{\phi}+\frac{\nabla_{\perp}{p_{i}}}{en}\right]\right.$
$\displaystyle\quad\left.+\frac{nc}{\Omega_{i}B}v_{\parallel
i}\nabla_{\parallel}\left(\nabla_{\perp}{\phi}+\frac{\nabla_{\perp}{p_{i}}}{en}\right)\right\\}+\mathcal{D}_{\omega}$
$\displaystyle\frac{d^{e}v_{\parallel e}}{dt}$
$\displaystyle=\frac{1}{m_{e}}\left(e\nabla_{\parallel}\phi-\frac{\nabla_{\parallel}p_{e}}{n}-0.71\nabla_{\parallel}T_{e}+e\eta_{\parallel}j_{\parallel}\right)$
(22)
$\displaystyle\quad+\frac{2}{3}\frac{\nabla_{\parallel}G_{e}}{n}+\frac{2cT_{e}}{eB}{C}_{\left({v_{\parallel
e}}\right)}+S_{\mathcal{M}\parallel e}+\mathcal{D}_{v_{\parallel e}}$
$\displaystyle\frac{d^{i}v_{\parallel i}}{dt}$
$\displaystyle=\frac{1}{m_{i}}\left(-e\nabla_{\parallel}\phi-\frac{\nabla_{\parallel}p_{i}}{n}+0.71\nabla_{\parallel}T_{e}-e\eta_{\parallel}j_{\parallel}\right)$
(23)
$\displaystyle+\frac{2T_{e}}{3n}\frac{\nabla_{\parallel}G_{i}}{n}-\frac{2cT_{i}}{eB}{C}_{\left({v_{\parallel
i}}\right)}+S_{\mathcal{M}\parallel i}+\mathcal{D}_{v_{\parallel i}}$
$\displaystyle\frac{d^{e}T_{e}}{dt}=\frac{2T_{e}}{3n}\left[\frac{d^{e}n}{dt}+\frac{1}{T_{e}}\nabla_{\parallel}\kappa^{e}_{\parallel}\nabla_{\parallel}T_{e}+\frac{5n}{m_{e}\Omega_{e}}{C}_{\left({T_{e}}\right)}\right.$
(24)
$\displaystyle\left.+\eta_{\parallel}\frac{j_{\parallel}^{2}}{T_{e}}+\frac{0.71}{e}(\nabla_{\parallel}j_{\parallel}-\frac{j_{\parallel}}{T_{e}}\nabla_{\parallel}T_{e})+\frac{1}{T_{e}}S_{E,e}\right]+\mathcal{D}_{T_{e}},$
$\displaystyle\frac{d^{i}T_{i}}{dt}$
$\displaystyle=\frac{2T_{i}}{3n}\left[\frac{d^{i}n}{dt}+\frac{1}{T_{i}}\nabla_{\parallel}\kappa^{i}_{\parallel}\nabla_{\parallel}T_{i}\right.$
(25)
$\displaystyle\quad\left.-\frac{5n}{m_{i}\Omega_{i}}{C}_{\left({T_{i}}\right)}+\frac{1}{T_{i}}S_{E,i}\right]+\mathcal{D}_{T_{i}}$

whereby the field-aligned electric current density is
$j_{\parallel}=en\left(v_{\parallel i}-v_{\parallel e}\right)$, the stress
tensor’s gyroviscous terms with some FLR effects contain
$G_{s}=\eta^{s}_{0}\left\\{2\nabla_{\parallel}v_{\parallel
s}+c\left[{C}_{\left({\phi}\right)}+{C}_{\left({p_{s}}\right)}/(q_{s}n)\right]\right\\}$,
and $\eta^{s}_{0}$ is the viscosity. The convective derivatives are
$d^{s}f/dt=\partial_{t}f+(c/B)\left[\phi,f\right]+v_{\parallel
s}\nabla_{\parallel}f$ with $\left[F,G\right]=\mathbf{b_{0}}\times\nabla
F\cdot\nabla G$ and $\mathbf{b_{0}}$ representing a constant unit vector
parallel to the unperturbed magnetic field. Consistent with the gyrokinetic
modelling, the field’s magnitude, $B$, decreases over the major radius of the
torus ($B\propto 1/R$), and $\mbox{\boldmath$\kappa$}=-{\bf{\hat{R}}}/R$.
Curvature drifts are encoded via diamagnetism in this fluid representation
Chapurin and Smolyakov (2016) with the curvature operator,
${C}_{\left({f}\right)}=\mathbf{b_{0}}\times\mbox{\boldmath$\kappa$}\cdot\nabla
f$, following (17) but with an unperturbed field vector distinguishing it from
electromagnetic theory. The constant coefficients $\kappa^{s}_{\parallel}$ and
$\eta^{s}_{\parallel}$ correspond to parallel thermal conductivity and
electrical resistivity, respectively. The 3-dimensional shearless field-
aligned coordinate system over which the drift-reduced Braginskii theory is
formulated in the physics-informed machine learning framework exactly matches
the gyrokinetic code’s geometry. The two-fluid model similarly consists of
deuterium ions and electrons with real masses (i.e. $m_{i}=3.34\times
10^{-27}\text{ kg}$ and $m_{e}=9.11\times 10^{-31}\text{ kg}$).

To consistently calculate the electric field response and integrate (20)–(25)
in time, classically one would compute the turbulent $\phi$ in drift-reduced
Braginskii theory by directly invoking quasineutrality and numerically solving
the boundary value problem given by the divergence-free condition of the
electric current density Scott (2003); Zholobenko  _et al._ (2021). For the
purposes of comparison with gyrokinetic modelling, we apply a novel technique
for computing the turbulent electric field consistent with the drift-reduced
Braginskii equations without explicitly evaluating $\nabla\cdot\mathbf{j}=0$
nor directly applying the Boussinesq approximation. Namely, this work applies
a validated physics-informed deep learning framework Mathews  _et al._ (2021a)
to infer the gauge-invariant $\phi$ directly from (20) and (24) for direct
analysis of electron pressure and electric field fluctuations in nonlinear
global electromagnetic gyrokinetic simulations and electrostatic two-fluid
theory. To review the pertinent aspects of the machine learning methodology
Mathews  _et al._ (2021a), every dynamical variable in equations (20)–(25) is
approximated by its own fully-connected neural network with local
spatiotemporal points $(x,y,t)$ from a reduced 2-dimensional domain as the
only inputs to the initial layer. The dynamical variable being represented by
each network is the sole output. In the middle of the architecture, every
network consists of 8 hidden layers with 50 neurons per hidden layer and
hyperbolic tangent activation functions all utilizing Xavier initialization
Glorot and Bengio (2010). The turbulent electric field consistent with drift-
reduced Braginskii theory is then learnt by optimizing the networks against
both $n_{e}$ and $T_{e}$ measurements along with the physical theory encoded
by (20) and (24). To be precise, as visualized in Figure 3 of Mathews  _et
al._ (2021a), learning proceeds by training the $n_{e}$ and $T_{e}$ networks
against the average $\mathcal{L}_{2}$-norm of their respective errors

$\mathcal{L}_{n_{e}}=\frac{1}{N_{0}}\sum_{i=1}^{N_{0}}\lvert
n^{*}_{e}(x^{i}_{0},y^{i}_{0},z^{i}_{0},t^{i}_{0})-n^{i}_{e,0}\rvert^{2},$
(26) $\mathcal{L}_{T_{e}}=\frac{1}{N_{0}}\sum_{i=1}^{N_{0}}\lvert
T^{*}_{e}(x^{i}_{0},y^{i}_{0},z^{i}_{0},t^{i}_{0})-T^{i}_{e,0}\rvert^{2},$
(27)

where
$\\{x_{0}^{i},y_{0}^{i},z_{0}^{i},t_{0}^{i},n_{e,0}^{i},T_{e,0}^{i}\\}^{N_{0}}_{i=1}$
correspond to the set of observed 2-dimensional data and the variables
$n^{*}_{e}$ and $T^{*}_{e}$ symbolize the electron density and temperature
predicted by the networks, respectively. Properties of all other dynamical
variables in the 6-field turbulence model are unknown. The only existing
requirement on the observational data in this deep learning framework is that
the measurements’ spatial and temporal resolutions should be be finer than the
autocorrelation length and time, respectively, of the turbulence structures
being analyzed. This scale condition is well-satisfied for the low-$\beta$
case and marginally-satisfied for the high-$\beta$ data analyzed. Model loss
functions are synchronously learnt by recasting (20) and (24) in the following
implicit form

$\displaystyle f_{n_{e}}$
$\displaystyle\coloneqq-\frac{d^{e}n}{dt}-\frac{2c}{B}\left[n{C}_{\left({\phi}\right)}-\frac{1}{e}{C}_{\left({p_{e}}\right)}\right]$
(28) $\displaystyle\quad-n\nabla_{\parallel}v_{\parallel
e}+S_{n}+\mathcal{D}_{n},$

$\displaystyle f_{T_{e}}$
$\displaystyle\coloneqq-\frac{d^{e}T_{e}}{dt}+\frac{2T_{e}}{3n}\left[\frac{d^{e}n}{dt}+\frac{1}{T_{e}}\nabla_{\parallel}\kappa^{e}_{\parallel}\nabla_{\parallel}T_{e}\right.$
(29)
$\displaystyle\left.+\frac{5n}{m_{e}\Omega_{e}}{C}_{\left({T_{e}}\right)}+\eta_{\parallel}\frac{j_{\parallel}^{2}}{T_{e}}+\frac{0.71}{e}(\nabla_{\parallel}j_{\parallel}\right.$
$\displaystyle\left.-\frac{j_{\parallel}}{T_{e}}\nabla_{\parallel}T_{e})+\frac{1}{T_{e}}S_{E,e}\right]+\mathcal{D}_{T_{e}},$

and then further normalized into dimensionless form Francisquez  _et al._
(2020a); Mathews  _et al._ (2021a). These unitless evolution equations of
$n_{e}$ and $T_{e}$ from the two-fluid theory are jointly optimized using

$\mathcal{L}_{f_{n_{e}}}=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}\lvert
f^{*}_{n_{e}}(x^{i}_{f},y^{i}_{f},z^{i}_{f},t^{i}_{f})\rvert^{2},$ (30)
$\mathcal{L}_{f_{T_{e}}}=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}\lvert
f^{*}_{T_{e}}(x^{i}_{f},y^{i}_{f},z^{i}_{f},t^{i}_{f})\rvert^{2},$ (31)

where $\\{x_{f}^{i},y_{f}^{i},z_{f}^{i},t_{f}^{i}\\}^{N_{f}}_{i=1}$ denote the
set of collocation points, and $f^{*}_{n_{e}}$ and $f^{*}_{T_{e}}$ are the
null partial differential equations prescribed by (28) and (29) in normalized
form directly evaluated by the neural networks. The set of collocation points
over which the partial differential equations are evaluated correspond to the
positions of the observed electron pressure data, i.e.
$\\{x_{0}^{i},y_{0}^{i},z_{0}^{i},t_{0}^{i}\\}^{N_{0}}_{i=1}=\\{x_{f}^{i},y_{f}^{i},z_{f}^{i},t_{f}^{i}\\}^{N_{f}}_{i=1}$.
This physics-informed framework is consequently a truly deep approximation of
drift-reduced Braginskii theory since every dynamical variable is represented
by its own fully-connected deep network which is then further differentiated
to build the cumulative computation graph reflecting every single term in the
evolution equations. Loss functions are optimized with mini-batch sampling
where $N_{0}=N_{f}=1000$ using L-BFGS—a quasi-Newton optimization algorithm
Liu and Nocedal (1989)—for 20 hours over 32 cores on Intel Haswell-EP
processors. We highlight that the only locally observed dynamical quantities
in these equations are 2-dimensional views of $n_{e}$ and $T_{e}$ (to emulate
gas puff imaging Mathews  _et al._ (2021b)) without any explicit information
about boundary conditions nor initializations nor ion temperature dynamics nor
parallel flows. This machine learning framework’s collocation grid consists of
a continuous spatiotemporal domain without time-stepping nor finite difference
schema in contrast with standard numerical codes. All analytic terms encoded
in these equations including high-order operators are computed exactly by the
neural networks without discretization. In that sense, it is a potentially
higher fidelity continuous representation of the continuum equations. While
the linearized polarization density—analogous to the Boussinesq
approximation—is employed in the gyrokinetic simulations, no such
approximations are explicitly applied by our physics-informed neural networks.

With sparse availability of measurements in fusion experiments, designing
diagnostic techniques for validating turbulence theories with limited
information is crucial. On this point, we note that our framework can
potentially be adapted to experimental measurements of electron pressure
Griener  _et al._ (2020); Bowman  _et al._ (2020); Mathews and Hughes (2021);
Mathews  _et al._ (2021b). To handle the particular case of 2-dimensional
turbulence data, we essentially assume slow variation of dynamics in the
$z$-coordinate and effectively set all parallel derivatives to zero. In
computational theory comparisons where no such limitations exist, training on
$n_{e}$ and $T_{e}$ in 3-dimensional space over long macroscopic timescales
can be easily performed via segmentation of the domain and parallelization,
but a limited spatial view away from numerical boundaries with reduced
dimensionality is taken to mirror experimental conditions for field-aligned
observations Zweben  _et al._ (2017) and fundamentally test what information
is indispensable to compare kinetic and fluid turbulence. To compare the
gyrokinetic and two-fluid theories as directly as possible, we analyze the
toroidal simulations at $z=L_{z}/3$ where no applied sources are present. We
further remark that, beyond the inclusion of appropriate collisional drifts
and sources, this technique is quite robust and generalizable to boundary
plasmas with multiple ions and impurities present due to the quasi-neutrality
assumptions underlying the two-fluid theory Poulsen  _et al._ (2020). This
turbulence diagnostic analysis technique is hence easily transferable, which
permits its systematic application across magnetic confinement fusion
experiments where the underlying physical model governing turbulent transport
is consistent. The framework sketched can thus be extended to different
settings especially in the interdisciplinary study (both numerical and
experimental) of magnetized collisional plasmas in space propulsion systems
and fusion energy and astrophysical environments if sufficient observations
exist. A full description and testing of the overall deep learning technique
is provided in Mathews  _et al._ (2021a).

##  IV Quantifying plasma turbulence consistency

A defining characteristic of a nonlinear theory is how it mathematically
connects dynamical variables. The focus of this work is quantitatively
examining the nonlinear relationship extant in plasma turbulence models
between electron pressure and electric field fluctuations. As outlined in
Section III, using a custom physics-informed deep learning framework whereby
the drift-reduced Braginskii equations are embedded in the neural networks via
constraints in the form of implicit partial differential equations, we
demonstrate for the first time direct comparisons of instantaneous turbulent
fields between electrostatic two-fluid theory and electromagnetic long-
wavelength gyrokinetic modelling in low-$\beta$ helical plasmas with results
visualized in Figure 1. This multi-network physics-informed deep learning
framework enables direct comparison of drift-reduced Braginskii theory with
gyrokinetics and bypasses demanding limitations in standard forward modelling
numerical codes which must precisely align initial conditions, physical
sources, numerical diffusion, and boundary constraints (e.g. particle and heat
fluxes into walls) in both gyrokinetic and fluid representations when
classically attempting comparisons of turbulence simulations and statistics.
Further, the theoretical and numerical conservation properties of these
simulations ordinarily need to be evaluated which can be a significant
challenge especially when employing disparate numerical methods with differing
discretization schema and integration accuracy altogether Francisquez  _et
al._ (2020a). We are able to overcome these hurdles by training on partial
electron pressure observations simultaneously with the plasma theory sought
for comparison. We specifically prove that the turbulent electric field
predicted by drift-reduced Braginskii two-fluid theory is largely consistent
with long-wavelength gyrokinetic modelling in low-$\beta$ helical plasmas.
This is also evident if analyzing $y$-averaged radial electric field
fluctuations and accounting for the inherent scatter ($\sigma_{PINN}$) from
the stochastic optimization employed as displayed in Figure 2. To clarify the
origins of $\sigma_{PINN}$, every time our physics-informed deep learning
framework is trained from scratch against the electron pressure observations,
the learned turbulent electric field will be slightly different due to the
random initialization of weights and biases for the networks along with mini-
batch sampling during training. To account for this stochasticity, we have
trained our framework anew 100 times while collecting the predicted turbulent
$E_{r}$ after each optimization. the quantity $\sigma_{PINN}$ corresponds to
the standard deviation from this collection. The two-fluid model’s results
displayed in Figure 2 is thus based upon computing $\langle E_{r}\rangle_{y}$
from 100 independently-trained physics-informed machine learning frameworks.
Their turbulent outputs are averaged together to produce the mean, while the
scatter associated with the 100 realizations—which approximately follows a
normal distribution—comprises the shaded uncertainty interval spanning 2
standard deviations. As visualized, the $\langle E_{r}\rangle_{y}$ profiles
predicted by the the electromagnetic gyrokinetic simulation and electrostatic
drift-reduced Braginskii model are generally in agreement within error bounds
at low-$\beta$.

Figure 1: The turbulent electric potential, $\phi$ (a gauge-invariant quantity
which is equivalent up to a scalar constant offset), and radial electric
field, $E_{r}$, concomitant with electron pressure fluctuations as predicted
by electrostatic drift-reduced Braginskii theory and electromagnetic
gyrokinetic modelling in low-$\beta$ conditions are in good quantitative
agreement. The two-fluid theory’s $\phi$ and $E_{r}$ are based upon the
training the physics-informed deep learning framework while the gyrokinetic
results are from the discontinuous Galerkin numerical solver. Figure 2: The
$y$-averaged turbulent radial electric field, $\langle E_{r}\rangle_{y}$, as
predicted by electrostatic drift-reduced Braginskii theory and electromagnetic
gyrokinetic modelling at low-$\beta$. The results plotted for the drift-
reduced Braginskii output here are based upon collecting 100 independently-
trained physics-informed neural networks. Figure 3: The turbulent electric
potential, $\phi$ (a gauge-invariant quantity which is equivalent up to a
scalar constant offset), and radial electric field, $E_{r}$, concomitant with
electron pressure fluctuations as predicted by electrostatic drift-reduced
Braginskii theory and electromagnetic gyrokinetic modelling in high-$\beta$
conditions are quantitatively inconsistent. The two-fluid theory’s $\phi$ and
$E_{r}$ are based upon the training of the physics-informed deep learning
framework while the gyrokinetic results are from the discontinuous Galerkin
numerical solver. Figure 4: The $y$-averaged turbulent radial electric field,
$\langle E_{r}\rangle_{y}$, as predicted by electrostatic drift-reduced
Braginskii theory and electromagnetic gyrokinetic modelling at high-$\beta$.
The results plotted for the drift-reduced Braginskii output here are based
upon collecting 100 independently-trained physics-informed neural networks.

In high-$\beta$ conditions where the particle source is artificially increased
by 10$\times$, electromagnetic effects become important and the electrostatic
two-fluid theory is markedly errant juxtaposed with electromagnetic
gyrokinetic simulations when considering the comparison in Figure 3. As
remarked above, multiple realizations are conducted to analyze the sample
statistics of the learned turbulent fields consistent with drift-reduced
Braginskii two-fluid theory based solely upon the intrinsic scatter during
training to account for the stochastic nature of the optimization. By
collecting 100 independently-trained realizations, the inherent uncertainty
linked to this intrinsic scatter can be evaluated as demonstrated in Figure 4.
These discrepancies indicate that the fluctuations in $\bf B_{\perp}$, which
are evaluated by solving the parallel Ampère equation (or, equivalently,
generalized Ohm’s law), cannot be neglected when considering plasma transport
across the inhomogeneous background magnetic field as in electrostatic theory.
While the fluid approximation is generally expected to be increasingly
accurate at high density due to strong coupling between electrons and ions,
these results underline the importance of electromagnetic effects even in
shear-free high-$\beta$ plasmas as found in planetary magnetospheres and
fusion experiments (e.g. dipole confinement Garnier  _et al._ (2006)), and for
the first time enables the degree of error between instantaneous fluctuations
to be precisely quantified across turbulence models. Uncertainty estimates
stemming from the stochastic framework in both regimes are reflected in Figure
5.

One should note that there are novel and different levels of errors to be
mindful of in this evaluation. For example, poor convergence arising from
nonuniqueness of the turbulent $E_{r}$ found during optimization against the
drift-reduced Braginskii equations, or $\mathcal{L}_{f_{n_{e}}}$ and
$\mathcal{L}_{f_{T_{e}}}$ remaining non-zero (and not below machine precision)
even after training Mathews  _et al._ (2021a). These potential errors exist on
top of standard approximations in the discontinuous Galerkin numerical scheme
representing the underlying gyrokinetic theory such as the implemented
Dougherty collision operator. Notwithstanding, when comparing the
electrostatic drift-reduced Braginskii theory to electromagnetic long-
wavelength gyrokinetic simulations at low-$\beta$, the results represent good
consistency in the turbulent electric field and all observed discrepancies are
mostly within the stochastic optimization’s expected underlying scatter.
Alternatively, when analyzing high-$\beta$ conditions, we observe that the
electrostatic two-fluid model cannot explain the turbulent electric field in
the electromagnetic gyrokinetic simulations. In particular, $\Delta
E_{r}\gtrsim 15\sigma_{PINN}$ in the bottom plot of Figure 5, where $\Delta
E_{r}$ is the difference in the instantaneous $E_{r}$ predicted by two-fluid
theory and gyrokinetics. This signals that the two models’ turbulent $E_{r}$
fluctuations are incompatible at high-$\beta$ conditions while quantifying the
separation.

Our multi-network physics-informed deep learning framework demonstrates the
suitability of electrostatic two-fluid theory as a good approximation of
turbulent electric fields in modern gyrokinetic simulations for low-$\beta$
helical plasmas with sufficient initial and boundary conditions. Conversely,
the electrostatic turbulence model is demonstrably insufficient for
high-$\beta$ plasmas. This finding is indicative of the importance of
including electromagnetic effects such as magnetic flutter in determining
cross-field transport even at $\beta_{e}\sim 2\%$. But field line
perturbations and the reduced numerical representation of gyrokinetic theory
are not the only effects at play causing mismatch at high-$\beta$: due to the
artificial nature of the strong localized particle source in $z$-space to
produce high-$\beta$ conditions, parallel dynamics including electron flows
along field lines and Ohmic heating effects become increasingly important.
These variables should now be observed or learnt first from the 2-dimensional
electron pressure measurements to accurately reconstruct the turbulent
electric field which signifies a departure from the expected low-$\beta$ edge
of tokamaks. Going forward, consideration of advanced magnetic geometries with
squeezing and shearing of fusion plasmas near null-points, which may even
couple-decouple upstream turbulence in tokamaks, will be important in the
global validation of reduced turbulence models with realistic shaping Ryutov
_et al._ (2008); Kuang, Labombard, and Brunner (2018); Kuang (2019); Nespoli
_et al._ (2020); Myra  _et al._ (2020). Also, while there is generally good
convergence in the low-$\beta$ gyrokinetic simulations at parameters relevant
to the edge of NSTX, numerical convergence in the artificially elevated
high-$\beta$ case is currently questionable and a potential source of
discrepancy. Running the high-$\beta$ gyrokinetic simulation with proper
collision frequency at an improved resolution of
$(N_{x},N_{y},N_{z},N_{v_{\parallel}},N_{\mu})=(48,96,18,10,5)$ would cost
roughly 4 million CPU-hours to check and we must leave such investigations for
future analysis.

Figure 5: The relative error in the instantaneous radial electric field
fluctuations ($\Delta E_{r}/\sigma_{PINN}$) between electrostatic drift-
reduced Braginskii theory and electromagnetic long-wavelength gyrokinetic
modelling is displayed in low-$\beta$ (top) and high-$\beta$ (bottom)
conditions. While all errors in the low-$\beta$ scenario are generally within
approximately 3–4 standard deviations and representative of mostly good
quantitative agreement, one must go to over $15\sigma_{PINN}$ to fully account
for the turbulent fields using an electrostatic two-fluid theory at
$\beta_{e}\sim 2\%$. The results are based upon collecting 100 independently-
trained physics-informed neural networks to compute the turbulent $E_{r}$ and
the intrinsic scatter in these predictions.

As for distinctiveness, we point out that the techniques used for computing
the displayed turbulent electric fields in the two cases are markedly
different. In particular, the long-wavelength gyrokinetic Poisson equation,
which is originally derived from the divergence-free condition of the electric
current density, is employed in the gyrokinetic simulations. In contrast,
simply the electron fluid evolution equations are used to infer the unknown
turbulent field fluctuations consistent with drift-reduced Braginskii theory
Mathews  _et al._ (2021a). A principle underlying these models is
quasineutrality, but this condition is not sufficient on its own. If one were
to apply equilibrium models such as the Boltzmann relation or simple ion
pressure balance as expected neoclassically, the turbulent electric field
estimates for these nonequilibrium plasmas with nontrivial cross-field
transport would be highly inaccurate Mathews  _et al._ (2021a). Further, no
external knowledge of boundary conditions such as sheath effects are
explicitly provided to the physics-informed deep learning framework, but this
information implicitly propagates from the walls into the observations of
electron pressure. This novel approach resultantly permits using limited 2D
measurements to compare a global 3D fluid turbulence model directly against 5D
gyrokinetics beyond statistical considerations for characterizing non-
diffusive intermittent edge transport Naulin (2007). All in all, the agreement
between drift-reduced Braginskii theory and gyrokinetics supports its usage
for predicting turbulent transport in low-$\beta$ shearless toroidal plasmas,
but an analysis of all dynamical variables is required for the full validation
of drift-reduced Braginskii theory. Alternatively, when 2-dimensional
experimental electron pressure measurements are available Furno  _et al._
(2008); Mathews  _et al._ (2021b), this technique can be used to infer $E_{r}$
and the resulting structure of turbulent fluxes heading downstream. And, if
independently diagnosed measurements of the laboratory plasma’s turbulent
$E_{r}$ exist, the overall turbulence theory can be expressly tested in
experiment.

##  V Conclusion

To probe the fundamental question of how similar two distinct turbulence
models truly are, using a novel technique to analyze electron pressure
fluctuations, we have directly demonstrated that there is good agreement in
the turbulent electric fields predicted by electrostatic drift-reduced
Braginskii theory and electromagnetic long-wavelength gyrokinetic simulations
in low-$\beta$ helical plasmas. As $\beta_{e}$ is increased to approximately
$2\%$, the 2-dimensional electrostatic nature of the utilized fluid theory
becomes insufficient to explain the microinstability-induced particle and heat
transport. Overall, by analyzing the interconnection between dynamical
variables in these global full-$f$ models, physics-informed deep learning can
quantitatively examine this defining nonlinear characteristic of turbulent
physics. In particular, we can now unambiguously discern when agreement exists
between multi-field turbulence theories and identify disagreement when
incompatibilities exist with just 2-dimensional electron pressure
measurements. This machine learning tool can therefore act as a necessary
condition to diagnose when reduced turbulence models are unsuitable, or,
conversely, iteratively construct and test theories till agreement is found
with observations.

While we focus on the electric field response to electron pressure in this
work, extending the analysis to all dynamical variables (e.g.
$T_{i},v_{\parallel e},v_{\parallel i}$) for full validation of reduced multi-
field turbulence models in a variety of regimes (e.g. collisionality, $\beta$,
closed flux surfaces with sheared magnetic field lines) using electromagnetic
fluid theory is the subject of future work. Also, since plasma fluctuations
can now be directly compared across models, as gyrokinetic codes begin
including fully kinetic neutrals, this optimization technique can help
validate reduced source models to accurately account for atomic and molecular
interactions with plasma turbulence in the edge of fusion reactors Thrysøe
_et al._ (2018, 2020) since these processes (e.g. ionization, recombination)
affect the local electric field. Further progress in the gyrokinetic
simulations such as the improved treatment of gyro-averaging effects Brizard
and Hahm (2007), collision operators Francisquez  _et al._ (2020b), and
advanced geometries Mandell (2021) will enable better testing and discovery of
hybrid reduced theories as well Zhu  _et al._ (2021). For example, in diverted
reactor configurations, electromagnetic effects become increasingly important
for transport near X-points where $\beta_{p}\rightarrow\infty$. A breakdown of
Alfvén’s theorem in these regions can also arise due to the impact of Coulomb
collisions and magnetic shear contributing to an enhanced perpendicular
resistivity Myra  _et al._ (2000) which presents an important test case of
non-ideal effects within reduced turbulence models. While our work supports
the usage of electrostatic two-fluid modelling, with adequate initial and
boundary conditions, over long-wavelength gyrokinetics for low-$\beta$
magnetized plasmas without magnetic shear, a comparison of all dynamical
variables beyond the turbulent electric field is required for a full
validation of the reduced model. Further investigations into reactor
conditions may suggest the modularization of individually validated fluid-
kinetic turbulence models across different regions in integrated global device
modelling efforts Hakim  _et al._ (2019); Merlo  _et al._ (2021). This task
can now be efficiently tackled through pathways in deep learning as
demonstrated by this new class of validation techniques. In addition,
precisely understanding the fundamental factors–both physical and
numerical–determining the prediction interval, $\sigma_{PINN}$, is the subject
of ongoing research in analyzing the nature (e.g. uniqueness, smoothness) of
chaotic solutions to fluid turbulence equations and the chaotic convergence
properties of physics-informed neural networks.

###### Acknowledgements.

The authors wish to thank A.Q. Kuang, D.R. Hatch, and G.W. Hammett for
insights shared and helpful discussions. Gyrokinetic simulations with the
Gkeyll code (https://gkeyll.readthedocs.io/en/latest/) were performed on the
Perseus cluster at Princeton University and the Cori cluster at NERSC. All
further analysis and codes run were conducted on MIT’s Engaging cluster and we
are grateful for assistance with computing resources provided by all teams.
The work is supported by the Natural Sciences and Engineering Research Council
of Canada (NSERC) by the doctoral postgraduate scholarship (PGS D), Manson
Benedict Fellowship, and the U.S. Department of Energy (DOE) Office of Science
under the Fusion Energy Sciences program by contracts DE-SC0014264, DE-
SC0014664, DE-AC02-09CH11466 via the Scientific Discovery through Advanced
Computing (SciDAC) program, and DE-AR0001263 via the Advanced Research
Projects Agency - Energy (ARPA-E) program.

## Data Availability

All relevant input files and codes to reproduce the data and results presented
in this work can be found on Github
https://github.com/AbhilashMathews/PlasmaPINNtheory ;
https://github.com/ammarhakim/gkyl-paper-
inp/tree/master/2021_PoP_GK_Fluid_PINN .

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# Comment on ‘Local available quantum correlations for Bell Diagonal states
and markovian decoherence’

David M. Bellorin R  Departamento de Física, Universidad Simón Bolívar, AP
89000, Caracas 1080, Venezuela.  Hermann L. Albrecht Q<EMAIL_ADDRESS>Departamento de Física, Universidad Simón Bolívar, AP 89000, Caracas 1080,
Venezuela.

###### Abstract

In this brief paper, we complete the analysis presented previously in [RMF 64
(2018) 662–670] regarding the quantifiers of the classical correlations and
the so-called local available quantum correlations for Bell Diagonal states. A
correction is introduced in their previous expressions once two cases within
the optimizations are included.

Quantum correlations, Quantum discord, Entanglement, Bell Diagonal states,
Werner states.

###### pacs:

03.65.Ud, 03.67.-a, 03.67.Mn.

††preprint: SB/F/493-21

##  I Introduction

In LAQC_BD , the analytical results of the correlation quantifiers related to
the so-called local available quantum correlations (LAQC) LAQC  for the family
of Bell Diagonal states Horodecki-BD_states  were presented. These states are
written in the Bloch representation as

$\rho^{BD}=\frac{1}{4}\left(\mathbbm{1}_{4}+\sum_{i=1}^{3}c_{i}\sigma_{i}\otimes\sigma_{i}\right),$
(1)

where the coefficients $c_{i}\in[-1,1]$ are such that $\rho^{BD}$ is a well-
behaved density matrix (i. e. has non-negative eigenvalues) and $\sigma_{i}$
are the well-known Pauli matrices, i.e.
$\sigma_{1}=\smallmatrixquantity(0&1\\\ 1&0)$,
$\sigma_{2}=\smallmatrixquantity(0&-i\\\ i&0)$, and
$\sigma_{3}=\smallmatrixquantity(1&0\\\ 0&-1)$. As is readily seen from (1),
BD states have null local Bloch vectors, and their reduced matrices
$\rho_{A(B)}$ are maximally mixed. That is, they are proportional to identity,

$\rho_{A}=\rho_{B}=\frac{1}{2}\,\mathbbm{1}_{2}.$ (2)

In their 2015 paper, Mundarain and Ladrón de Guevara introduced a new type of
quantum correlations defined in the complementary basis of an optimal
computational one. To do so, they establish a generic orthonormal basis

$\displaystyle\ket{\mu_{0}^{(n)}}$ $\displaystyle=$
$\displaystyle\cos\quantity(\frac{\theta_{n}}{2})\ket{0^{(n)}}+\sin\quantity(\frac{\theta_{n}}{2})\exp^{i\phi_{n}}\ket{1^{(n)}},$
(3) $\displaystyle\ket{\mu_{1}^{(n)}}$ $\displaystyle=$
$\displaystyle-\sin\quantity(\frac{\theta_{n}}{2})\ket{0^{(n)}}+\cos\quantity(\frac{\theta_{n}}{2})\exp^{i\phi_{n}}\ket{1^{(n)}},$

where $n=1$ denotes subsystem A and $n=2$ subsystem B, respectively. The
optimal basis is fixed by minimizing the conditional entropy

$S\quantity(\rho_{AB}||X_{\rho})=\min_{\Omega_{c}}S\quantity(\rho_{AB}||\chi_{\rho})\,,$
(4)

where
$S\quantity(\rho_{AB}||\chi)=-\mathrm{Tr}(\rho\mathrm{log}_{2}\chi)-S(\rho_{AB})$
is the relative entropy and

$\displaystyle\chi_{\rho}=$
$\displaystyle\sum_{ij}R_{ij}\,\outerproduct{\mu_{i}^{(1)},\mu_{j}^{(2)}}{\mu_{i}^{(1)},\mu_{j}^{(2)}}\,,$
(5) $\displaystyle R_{ij}=$
$\displaystyle\expectationvalue{\rho_{AB}}{\mu_{i}^{(1)},\mu_{j}^{(2)}}\,.$

The classical correlations quantifier defined in this context in LAQC  is
given by

$\mathcal{C}(\rho_{AB})\equiv
S\left(X_{\rho_{AB}}||\Pi_{{}_{X_{\rho_{AB}}}}\right),$ (6)

where $\Pi_{{}_{X_{\rho_{AB}}}}$ is the product state nearest to
$X_{\rho_{AB}}$.

As was shown by Modi et al. Modi-RelativeEntropy , the relative entropy in the
above expression can be written in terms of the Mutual Information, given by

$I(\rho_{AB})\equiv
S\quantity(\rho_{A})+S\quantity(\rho_{B})-S\quantity(\rho_{AB}),$ (7)

of state $X_{\rho_{AB}}$. Moreover, since the mutual information (7) may be
written as

$I\quantity(\rho_{AB})=\sum_{i,j}P(i_{A},j_{B})\log_{2}\quantity[\frac{P(i_{A},j_{B})}{P(i_{A})P(j_{B})}],$
(8)

where
$P(i_{A},j_{B})=\expectationvalue{\rho_{AB}}{\mu_{i}^{(1)},\mu_{j}^{(2)}}$ are
the probability distributions corresponding to $\rho_{AB}$, and
$P(i_{A})=\expectationvalue{\rho_{A}}{\mu_{i}^{(1)}}$,
$P(j_{B})=\expectationvalue{\rho_{B}}{\mu_{i}^{(2)}}$ the ones corresponding
to its marginals $\rho_{A}$ and $\rho_{B}$, the classical correlations
quantifier (6) can be written in terms of the
$R_{ij}\quantity(\theta_{A},\phi_{A},\theta_{B},\phi_{B})$ coefficients as

$\displaystyle\mathcal{C}\quantity(\rho_{AB})$
$\displaystyle=\min_{\smallmatrixquantity{\theta_{A},\phi_{A}\\\
\theta_{B},\phi_{B}}}\Bigg{\\{}\sum_{i,j}R_{ij}\quantity(\theta_{A},\phi_{A},\theta_{B},\phi_{B})$
(9) $\displaystyle\mbox{\quad
\quad}\times\log_{2}\quantity[\frac{R_{ij}\quantity(\theta_{A},\phi_{A},\theta_{B},\phi_{B})}{R_{i}\quantity(\theta_{A},\phi_{A})R_{j}\quantity(\theta_{B},\phi_{B})}]\Bigg{\\}}.$

The above minimization over the angles $\theta_{A}$, $\phi_{A}$, $\theta_{B}$,
and $\phi_{B}$ defines a new computational basis
$\quantity{\ket{0^{(m)}}_{opt},\ket{1^{(m)}}_{opt}}$ relative to
$\quantity{\ket{\mu_{i}^{(n)}}}$ (3). The state $\rho_{AB}$ is rewritten, and
the complementary one is then defined in terms of a new general orthonormal
basis:

$\displaystyle\ket{u_{0}^{(m)}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\quantity(\ket{0^{(m)}}_{opt}+\exp^{i\Phi_{m}}\ket{1^{(m)}}_{opt})\,,$
$\displaystyle\ket{u_{1}^{(m)}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\quantity(\ket{0^{(m)}}_{opt}-\exp^{i\Phi_{m}}\ket{1^{(m)}}_{opt})\,.$
(10)

The corresponding probability distributions,

$P_{\Phi}(i_{A},j_{B},\Phi_{1},\Phi_{2})=\expectationvalue{\tilde{\rho}_{AB}}{u_{i}^{A},u_{j}^{B}},$
(11)

where $\tilde{\rho}_{AB}$ is the density matrix ${\rho}_{AB}$ written in the
optimal computational basis, and the marginal probability distributions are
determined. The maximization of $I\quantity(\Phi_{1},\Phi_{2})$ (8)
corresponds to the LAQC quantifier:

$\mathcal{L}(\rho_{AB})\equiv\max_{\quantity{\Phi_{1},\Phi_{2}}}I\quantity(\Phi_{1},\Phi_{2})\,.$
(12)

##  II Local available quantum correlations of Bell diagonal states

Since Bell diagonal (BD) states have null local Bloch vector, it is
straightforward that they are invariant under subsystem exchange
$\mathbf{A}\leftrightarrow{}\mathbf{B}$. Therefore, the parametrization of the
orthogonal basis $\quantity{\ket{\mu_{i}^{(n)}}}$ (3) needs to respect this
symmetry. As a consequence, only two angles, $\theta$ and $\phi$, are
necessary to parametrize (3). With this taken into account, the coefficients
$R_{ij}(\theta,\phi)$ are given by

$\displaystyle\begin{aligned}
R_{ij}(\theta,\phi)&=\frac{1}{4}\quantity[1+(-1)^{i+j}c_{3}]\\\
&+(-1)^{i+j}\frac{1}{2}\cos^{2}\quantity(\frac{\theta}{2})\sin^{2}\quantity(\frac{\theta}{2})\\\
&\times\quantity\big[(c_{1}+c_{2})+\cos(2\phi)(c_{1}-c_{2})-2c_{3}],\end{aligned}$
(13)

with $R_{00}(\theta,\phi)=R_{11}(\theta,\phi)$,
$R_{01}(\theta,\phi)=R_{10}(\theta,\phi)$, and $R_{i}=\frac{1}{2}$. The
minimization in (9) leads to three different cases:

  1. I.)

For $\theta=0$ and $\phi=0$:

$R_{00}(0,0)=\frac{1}{4}\quantity(1+c_{3}),\quad
R_{01}(0,0)=\frac{1}{4}\quantity(1-c_{3}).$ (14)

  2. II.)

For $\theta=\frac{\pi}{2}$ and $\phi=0$:

$R_{00}\quantity(\frac{\pi}{2},0)=\frac{1}{4}\quantity(1+c_{1}),\quad
R_{01}\quantity(\frac{\pi}{2},0)=\frac{1}{4}\quantity(1-c_{1}).$ (15)

  3. III.)

For $\theta=\frac{\pi}{2}$ and $\phi=\frac{\pi}{2}$:

$R_{00}\quantity(\frac{\pi}{2},\;\frac{\pi}{2})=\frac{1}{4}\quantity(1+c_{2}),R_{01}\quantity(\frac{\pi}{2},\frac{\pi}{2})=\frac{1}{4}\quantity(1-c_{2}).$
(16)

Therefore, by defining

$c_{m}\equiv\min\quantity\big{\absolutevalue{c_{1}},\absolutevalue{c_{2}},\absolutevalue{c_{3}}},$
(17)

we can write the classical correlations quantifier (9) as

$\displaystyle\mathcal{C}\quantity(\rho^{BD})=$
$\displaystyle\frac{1+c_{m}}{2}\log_{2}(1+c_{m})$ (18)
$\displaystyle+\frac{1-c_{m}}{2}\log_{2}(1-c_{m}).$

The above expression is the same as eq. (33) in LAQC_BD  but now the
minimization achieved for $\theta=\frac{\pi}{2}$ and $\phi=0$ when
$c_{m}=\absolutevalue{c_{1}}$ has been included.

To determine the LAQC quantifier (12), we have to rewrite the density matrix
$\rho^{BD}$ in the optimal computational basis. Contrary to what is stated in
LAQC_BD , the density matrix of BD states does not remain invariant when
written in the optimal computational basis. That is only true for Werner
Werner  and Werner-like states Werner-Like_Ghosh ; Werner-Like_Verstraete .

The density matrix $\tilde{\rho}^{BD}$ and their corresponding $P(i,j,\Phi)$
for each $\theta$ and $\phi$, with $P(0,0,\Phi)=P(1,1,\Phi)$,
$P(0,1,\Phi)=P(1,0,\Phi)$, and $P(i,\Phi)=\frac{1}{2}$, are the following:

  1. I.)

For $\theta=0$ and $\phi=0$:

$\tilde{\rho}^{BD}=\frac{1}{4}\matrixquantity(1+c_{3}&0&0&c_{1}-c_{2}\\\
0&1-c_{3}&c_{1}+c_{2}&0\\\ 0&c_{1}+c_{2}&1-c_{3}&0\\\
c_{1}-c_{2}&0&0&1+c_{3})$ (19)

and

$\displaystyle P(0,0,\Phi)$
$\displaystyle=\frac{1}{4}\quantity[1+\frac{c_{1}+c_{2}}{2}+\frac{c_{1}-c_{2}}{2}\cos(2\Phi)],$
(20) $\displaystyle P(1,0,\Phi)$
$\displaystyle=\frac{1}{4}\quantity[1-\frac{c_{1}+c_{2}}{2}-\frac{c_{1}-c_{2}}{2}\cos(2\Phi)].$

  2. II.)

For $\theta=\frac{\pi}{2}$ and $\phi=0$:

$\tilde{\rho}^{BD}=\frac{1}{4}\matrixquantity(1+c_{1}&0&0&c_{3}-c_{2}\\\
0&1-c_{1}&c_{3}+c_{2}&0\\\ 0&c_{3}+c_{2}&1-c_{1}&0\\\
c_{3}-c_{2}&0&0&1+c_{1})$ (21)

and

$\displaystyle P(0,0,\Phi)$
$\displaystyle=\frac{1}{4}\quantity[1+\frac{c_{3}+c_{2}}{2}+\frac{c_{3}-c_{2}}{2}\cos(2\Phi)],$
(22) $\displaystyle P(1,0,\Phi)$
$\displaystyle=\frac{1}{4}\quantity[1-\frac{c_{3}+c_{2}}{2}-\frac{c_{3}-c_{2}}{2}\cos(2\Phi)].$

  3. III.)

For $\theta=\frac{\pi}{2}$ and $\phi=\frac{\pi}{2}$:

$\tilde{\rho}^{BD}=\frac{1}{4}\matrixquantity(1+c_{2}&0&0&c_{3}-c_{1}\\\
0&1-c_{2}&c_{3}+c_{1}&0\\\ 0&c_{3}+c_{1}&1-c_{2}&0\\\
c_{3}-c_{1}&0&0&1+c_{2})$ (23)

and

$\displaystyle P(0,0,\Phi)$
$\displaystyle=\frac{1}{4}\quantity[1+\frac{c_{3}+c_{1}}{2}+\frac{c_{3}-c_{1}}{2}\cos(2\Phi)],$
(24) $\displaystyle P(1,0,\Phi)$
$\displaystyle=\frac{1}{4}\quantity[1-\frac{c_{3}+c_{1}}{2}-\frac{c_{3}-c_{1}}{2}\cos(2\Phi)].$

For each $\theta$ and $\phi$, $\Phi$ depends on
$\absolutevalue{c_{1}}>\absolutevalue{c_{2}}$,
$\absolutevalue{c_{2}}>\absolutevalue{c_{3}}$, or
$\absolutevalue{c_{1}}>\absolutevalue{c_{3}}$, respectively. Therefore, as was
done with the classical correlations quantifier (18), defining

$c_{M}\equiv\max\quantity{\absolutevalue{c_{1}},\absolutevalue{c_{2}},\absolutevalue{c_{3}}}$
(25)

allows us to write a general expression for the LAQC quantifier that
encompasses all these possibilities:

$\displaystyle\mathcal{L}\quantity(\rho^{BD})=$
$\displaystyle\frac{1+c_{M}}{2}\log_{2}(1+c_{M})$ (26)
$\displaystyle\mbox{\quad\quad}+\frac{1-c_{M}}{2}\log_{2}(1-c_{M}).$

As with the classical correlations quantifiers, the above expression is
equivalent to the one presented in eq. (36) of LAQC_BD . Nevertheless, this
newly defined $c_{M}$ also includes $\absolutevalue{c_{3}}$. The case of
$c_{M}=\absolutevalue{c_{3}}$ arises when the density matrix ${\rho}^{BD}$ is
written in the optimal computational basis with $\theta=\frac{\pi}{2}$.

##  III Conclusions

In this brief paper, we have completed the previous results regarding the so-
called local available quantum correlations (LAQC) for Bell diagonal (BD)
states. By including the cases of $\theta=\frac{\pi}{2}$ and $\phi=0$ in the
minimization for determining the classical correlations, as well as the
transformation of $\rho^{BD}$ when the optimal computational basis has
$\theta=\frac{\pi}{2}$ we extended the definitions of $c_{m}$ and $c_{M}$ to
include $\absolutevalue{c_{1}}$ and $\absolutevalue{c_{3}}$, respectively.

##  IV Acknowledgments

This work was partially funded by the _2020 BrainGain Venezuela_ grant awarded
to H. Albrecht by the _Physics without Frontiers_ program of the ICTP. The
authors would like to thank the support given by the research group GID-30,
_Teoría de Campos y Óptica Cuántica_ , at the Universidad Simón Bolívar,
Venezuela. The authors would like to also thank D. Mundarain, from the
Universidad Católica del Norte, Chile, as well as M.I. Caicedo and J.
Stephany, from Universidad Simón Bolívar, Venezuela, for their comments and
suggestions.

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