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\begin_body

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\tht}{\vartheta}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ph}{\varphi}
\end_inset


\begin_inset FormulaMacro
\newcommand{\balpha}{\boldsymbol{\alpha}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\btheta}{\boldsymbol{\theta}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bJ}{\boldsymbol{J}}
\end_inset


\begin_inset FormulaMacro
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\begin_inset FormulaMacro
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\begin_inset FormulaMacro
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\begin_inset FormulaMacro
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\begin_inset FormulaMacro
\newcommand{\pset}{{\bf p}}
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\end_layout

\begin_layout Title
Neoclassical Toroidal Viscous Torque
\end_layout

\begin_layout Author
Christopher Albert
\end_layout

\begin_layout Date
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
today
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Neoclassical toroidal viscous torque, often called neoclassical toroidal
 viscosity or NTV, is a result from non-axisymmetric perturbations in an
 originally axisymmetric plasma equilibrium of a tokamak.
 The underlying theory of neoclassical transport relies on distorted but
 still nested flux surfaces, i.e.
 
\emph on
non-resonant 
\emph default
magnetic perturbations.
 In case of 
\emph on
resonant
\emph default
 magnetic perturbations (RMPs) one must exclude resonant surfaces from the
 analysis that yield a different mechanism of 
\emph on
resonant torque
\emph default
.
 At sufficient distance from the resonant regions NTV theory is then applicable,
 and the overall torque is the sum of non-resonant NTV torque and resonant
 torque, respectively dominant in different radial regions of the plasma.
 Apart from additional resonant torque contributions an important point
 affecting the expected accuracy of NTV results is the reliance on a on
 a given perturbed plasma equilibrium for neoclassical computations.
 Finally, both, resonant and NTV torque affect this equilibrium, so it must
 be found in a self-consistent way.
 Such an effort including only NTV has been undertaken in
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "Park2017"
literal "false"

\end_inset

.
 A full treatment must however also include resonant torque, requiring a
 full Monte-Carlo kinetic computation coupled to Maxwell's equations necessary
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "Albert2016"
literal "false"

\end_inset

.
 Analysis of NTV is still useful, as it allows to study and quantify the
 influence of the non-resonant part of the perturbation on its own and is
 much less computationally intensive than the full kinetic treatment of
 perturbed equilibria.
\end_layout

\begin_layout Standard
Generally we speak of toroidal torque as the driving term affecting 
\emph on
kinematic 
\emph default
toroidal angular momentum, as opposed to the total toroidal momentum including
 electromagnetic effects that is always conserved.
 Apart from a possible kinematic momentum source term from neutral beam
 injection, internal torque affecting toroidal plasma rotation is always
 electromagnetic, i.e.
 linked to the Lorentz 
\begin_inset Formula $\v J\times\v B$
\end_inset

 force.
 Within ideal MHD equilibria, currents 
\begin_inset Formula $\v J$
\end_inset

 and magnetic field 
\begin_inset Formula $\v B$
\end_inset

 are parallel to flux surfaces and do not produce any forces, since radial
 Lorentz forces balanced by 
\begin_inset Formula $\nabla p$
\end_inset

 via
\begin_inset Formula 
\begin{equation}
\nabla p=\v J\times\v B.
\end{equation}

\end_inset

The MHD force balance is also valid in non-axisymmetric fields, where ideal
 treatment is only possible away from resonant regions where surfaces of
 
\begin_inset Formula $p=p_{0}+\delta p=\mathrm{const.}$
\end_inset

 coincide with perturbed flux surfaces 
\begin_inset Formula $r=r_{0}+\delta r=\mathrm{const}$
\end_inset

.
 When taking neoclassical transport into account one can show that it is
 radially 
\emph on
ambipolar
\emph default
 in axisymmetric devices to leading order, i.e.
 there is no net radial charge transport by different species.
 This changes in a perturbed configuration, where the transport across the
 perturbed flux surfaces (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Perturbed-and-unperturbed"

\end_inset

) becomes 
\emph on
non-ambipolar
\emph default
 and generates a net radial current.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{pertunpert.tpx}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Perturbed and unperturbed flux surfaces.
 The projection of perturbed currents 
\begin_inset Formula $J^{r_{0}}=\v J\cdot\nabla r_{0}$
\end_inset

 to the original radial direction 
\begin_inset Formula $\nabla r_{0}$
\end_inset

 is of the order one and periodic in flux surface angles.
 For NTV we are rather interested in the small 
\begin_inset Formula $\delta J^{r}=\v J\cdot\nabla r$
\end_inset

 produced by non-ambipolar transport across perturbed flux surfaces 
\begin_inset Formula $r=\mathrm{const.}$
\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:Perturbed-and-unperturbed"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The resulting toroidal torque density is defined as the covariant component
 with respect to the toroidal angle 
\begin_inset Formula $\ph$
\end_inset

 parametrizing the 
\emph on
perturbed
\emph default
 flux surfaces,
\begin_inset Formula 
\begin{equation}
\pi_{\ph}=(\v J\times\v B)_{\ph}=\frac{1}{\sqrt{g}}\varepsilon\delta J^{r}B^{\tht}.\label{eq:lag}
\end{equation}

\end_inset

Here we have marked 
\begin_inset Formula $\delta J^{r}=\v J\cdot\nabla r$
\end_inset

 as a small quantity resulting from 
\begin_inset Formula $\v J=\v J_{0}+\delta\v J$
\end_inset

.
 This is not to be confused with the torque with respect to 
\emph on
unperturbed
\emph default
 flux surfaces having different coordinates 
\begin_inset Formula $(r_{0},\tht_{0},\ph_{0})$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\pi_{\ph_{0}}=(\v J\times\v B)_{\ph_{0}}=\frac{1}{\sqrt{g}}(\delta J^{r_{0}}B^{\tht_{0}}-J^{\tht_{0}}\delta B^{r_{0}}).\label{eq:eul}
\end{equation}

\end_inset

Here also the radial component 
\begin_inset Formula $\delta B^{r_{0}}=\v B\cdot\nabla r_{0}$
\end_inset

 from 
\begin_inset Formula $\v B=\v B_{0}+\delta\v B$
\end_inset

 enters, where 
\begin_inset Formula $\v B_{0}\cdot\nabla r_{0}=0$
\end_inset

 on unperturbed flux surfaces.
 The contribution 
\begin_inset Formula 
\begin{equation}
J^{r_{0}}=\v J\cdot\nabla r_{0}=\v J\cdot\nabla(r-\delta r)=\delta J^{r}-\v J\cdot\nabla\delta r.
\end{equation}

\end_inset

now contains not only small currents from 
\begin_inset Formula $\delta J^{r}$
\end_inset

 non-ambipolar transport , but also currents parallel to perturbed flux
 surfaces, containing angular components of the overall (order of 
\begin_inset Formula $\v J_{0}$
\end_inset

) current 
\begin_inset Formula $\v J$
\end_inset

 that is not parallel to surfaces 
\begin_inset Formula $r_{0}=\mathrm{const.}$
\end_inset

 but rather to 
\begin_inset Formula $r=\mathrm{const.}$
\end_inset

 due to the distortion.
 Since the perturbation is periodic in angles, those additional periodic
 contribution cancel out when computing the flux-surface averaged radial
 torque density, and the two results become equivalant to leading order
 in the perturbation.
 In case of destroyed flux surfaces it is no longer possible to use Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lag"

\end_inset

 and one can only compute overall (including resonant) torque within Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:eul"

\end_inset

 and flux-surface averaging.
 In contrast, NTV torque corresponds to the toroidal component of the torque
 generate by radial currents of non-ambipolar neoclassical transport with
 respect to perturbed flux surfaces and can be most compactly formulated
 in the Lagrangian picture working 
\emph on
on 
\emph default
those surfaces in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lag"

\end_inset

.
 Later we can make use of the assumption that the separation of perturbed
 from unperturbed flux surfaces is infintesimal, and use flux coordinates
 and orbits with regard to the unperturbed surfaces while evaluating toroidal
 torque on perturbed surfaces.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Park2017"
literal "false"

\end_inset

J.-K.
 Park and N.
 C.
 Logan, “Self-consistent perturbed equilibrium with neoclassical toroidal
 torque in tokamaks,” Phys.
 Plasmas, vol.
 24, no.
 3, p.
 032505, Mar.
 2017.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Albert2016"
literal "false"

\end_inset

C.
 G.
 Albert, M.
 F.
 Heyn, S.
 V Kasilov, W.
 Kernbichler, A.
 F.
 Martitsch, and A.
 M.
 Runov, “Kinetic modeling of 3D equilibria in a tokamak,” J.
 Phys.
 Conf.
 Ser., vol.
 775, no.
 1, p.
 012001, Nov.
 2016.
\end_layout

\end_body
\end_document