rmps.lyx

#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\use_hyperref false
\papersize default
\use_geometry false
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\use_minted 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header

\begin_body

\begin_layout Section*
\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
\newcommand{\bGamma}{\boldsymbol{\Gamma}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bOmega}{\boldsymbol{\Omega}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\d}{\text{d}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\t}[1]{\text{#1}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\m}{\text{m}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\v}[1]{\boldsymbol{#1}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\de}[1]{\mathcal{#1}}
\end_inset


\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\renewcommand{\t}[1]{\mathbf{#1}}
\end_inset


\end_layout

\begin_layout Section*
MHD equilibrium in density form
\end_layout

\begin_layout Standard
MHD equilibrium equations in vectorial notation relating current density
 
\begin_inset Formula $\v J$
\end_inset

, magnetic flux density 
\begin_inset Formula $\v B$
\end_inset

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

 are the force balance equation
\begin_inset Formula 
\begin{equation}
\v J\times\v B=\nabla p
\end{equation}

\end_inset

and the magnetostatic limit Maxwell's equations - namely source-freeness
 of 
\begin_inset Formula $\v B$
\end_inset

, Ampère's law linking 
\begin_inset Formula $\v J$
\end_inset

 and 
\begin_inset Formula $\v H$
\end_inset

, and the linear constitutive relation, linking 
\begin_inset Formula $\v B$
\end_inset

 and 
\begin_inset Formula $\v H$
\end_inset

 by a given permeability tensor 
\begin_inset Formula $\hat{\mu}$
\end_inset

, equal to 
\begin_inset Formula $\mu_{0}$
\end_inset

 in a vacuum background,
\begin_inset Formula 
\begin{align}
\nabla\cdot\v B & =0,\label{eq:divfree}\\
\nabla\times\v H & =\v J,\\
\v H & =\hat{\mu}^{-1}\v B.
\end{align}

\end_inset

To fulfill divergence-freeness 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:divfree"

\end_inset

 often a vector potential 
\begin_inset Formula $\v A$
\end_inset

 is introduced, replacing 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:divfree"

\end_inset

 by 
\begin_inset Formula 
\begin{equation}
\v B=\nabla\times\v A.
\end{equation}

\end_inset

This makes the solution in 
\begin_inset Formula $\v A$
\end_inset

 non-unique up to gauge freedom of adding 
\begin_inset Formula $\nabla\chi$
\end_inset

 to it.
\end_layout

\begin_layout Standard
The natural way to represent the involved vector fields in curvilinear geometry
 is via covariant vector field components for 
\begin_inset Formula $\v A$
\end_inset

 and 
\begin_inset Formula $\v H$
\end_inset

, and contravariant vector density components for 
\begin_inset Formula $\v B$
\end_inset

 and 
\begin_inset Formula $\v J$
\end_inset

, and covariant vector density components for the force density 
\begin_inset Formula $\v F=\v J\times\v B=\nabla p$
\end_inset

 matching Lorentz force and thermodynamic force by the pressure gradient.
 Those are usual components weighted via the Jacobian 
\begin_inset Formula $\sqrt{g}$
\end_inset

 with
\begin_inset Formula 
\begin{align}
\de J^{k} & =\sqrt{g}J^{k},\\
\de B^{k} & =\sqrt{g}B^{k},\\
\mathcal{F}_{k} & =\sqrt{g}F_{k}=\sqrt{g}\partial_{k}p.
\end{align}

\end_inset

This way we obtain metric-free components of the Lorentz force
\begin_inset Formula 
\[
\mathcal{F}_{i}=\varepsilon_{ijk}\mathcal{J}^{j}\mathcal{B}^{k}
\]

\end_inset

and field equations
\begin_inset Formula 
\begin{align}
\partial_{i}\mathcal{B}^{i} & =0,\label{eq:divfree2}\\
\varepsilon^{ijk}\partial_{i}H_{j} & =\mathcal{J}^{k}.
\end{align}

\end_inset

Again, Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:divfree2"

\end_inset

 can be replaced by 
\begin_inset Formula 
\begin{equation}
\mathcal{B}^{i}=\varepsilon^{ijk}\partial_{j}A_{k},
\end{equation}

\end_inset

with 
\begin_inset Formula $A_{k}$
\end_inset

 replacing unknowns 
\begin_inset Formula $\mathcal{B}^{i}$
\end_inset

 up to gauge freedom.
 Equations involving the metric are relation to thermodynamic force and
 consitutive relation,
\begin_inset Formula 
\begin{align}
\mathcal{F}_{i} & =\sqrt{g}\partial_{i}p,\\
H_{i} & =\mu_{ij}\mathcal{B}^{j}.
\end{align}

\end_inset

The metric enters the permeability tensor density components 
\begin_inset Formula $\mu_{jk}$
\end_inset

.
 For a vacuum background we have
\begin_inset Formula 
\begin{equation}
H_{j}=\frac{g_{jk}}{\mu_{0}}B^{k}=\frac{g_{jk}}{\mu_{0}\sqrt{g}}\mathcal{B}^{k}.
\end{equation}

\end_inset

 
\end_layout

\begin_layout Section*
Flux coordinates
\end_layout

\begin_layout Standard
For axisymmetric equilibria, the Hamiltonian form of 
\begin_inset Formula $\v B$
\end_inset

 proves the existence of nested flux surfaces.
 We use flux-surface aligned coordinates 
\begin_inset Formula $(r,\tht,\ph)$
\end_inset

 with flux surfaces labelled by the radial coordinate 
\begin_inset Formula $r$
\end_inset

.
 Taking a scalar product of the force-balance equation by 
\begin_inset Formula $\v B$
\end_inset

 or 
\begin_inset Formula $\v J$
\end_inset

 yields the result that both vector fields have field-lines lying on surfaces
 of constant 
\begin_inset Formula $p=p(r)$
\end_inset

.
 Take unperturbed field to be straight in 
\begin_inset Formula $\tht,\ph$
\end_inset

 on flux surfaces 
\begin_inset Formula $r=\mathrm{const.}$
\end_inset

 In a metric-free representation, we use covariant components of the vector
 potential depending only on the radial coordinate
\series bold

\begin_inset Formula 
\begin{equation}
A_{r}=0,\quad A_{\tht}=A_{\tht}(r),\quad A_{\ph}=A_{\ph}(r).
\end{equation}

\end_inset


\series default
Taking the magnetic flux density 
\begin_inset Formula $\v B=\nabla\times\v A$
\end_inset

 yields contravariant density components depending only on 
\begin_inset Formula $r$
\end_inset

 with
\begin_inset Formula 
\begin{align}
\mathcal{B}^{r}(r) & =\sqrt{g}B^{r}=\varepsilon^{rjk}\partial_{j}A_{k}=0\\
\mathcal{B}^{\tht}(r) & =\sqrt{g}B^{\tht}=\varepsilon^{\tht jk}\partial_{j}A_{k}=-\frac{\d A_{\ph}(r)}{\d r},\\
\mathcal{B}^{\ph}(r) & =\sqrt{g}B^{\tht}=\varepsilon^{\ph jk}\partial_{j}A_{k}=\frac{\d A_{\tht}(r)}{\d r}.
\end{align}

\end_inset

The magnetic field is then represented in contravariant density form by
\begin_inset Formula 
\begin{align}
\v B(\v x) & =\sqrt{g(\v x)}^{-1}(\de B^{\tht}(r)\v e_{\tht}(\v x)+\de B^{\ph}(r)\v e_{\ph}(\v x))\nonumber \\
 & =\sqrt{g(\v x)}^{-1}\de B^{\ph}(r)(\iota(r)\v e_{\tht}(\v x)+\v e_{\ph}(\v x)).
\end{align}

\end_inset

with rotational transform
\begin_inset Formula 
\begin{equation}
\iota(r)=\frac{\de B^{\tht}(r)}{\de B^{\ph}(r)}=\frac{B^{\tht}(\v x)}{B^{\ph}(\v x)}.
\end{equation}

\end_inset

Covariant density components of the Lorentz force are
\begin_inset Formula 
\begin{align*}
\mathcal{F}_{r}(\v x) & =\mathcal{J}^{\tht}(\v x)\mathcal{B}^{\ph}(r)-\mathcal{J}^{\ph}(\v x)\mathcal{B}^{\tht}(r)\\
\mathcal{F}_{\tht}(\v x) & =0\\
\mathcal{F}_{\ph}(\v x) & =0
\end{align*}

\end_inset

where the metric enters the covariant thermodynamic force density components
 via the counteracting pressure
\begin_inset Formula 
\begin{equation}
\mathcal{F}_{r}(\v x)=\sqrt{g(\v x)}p^{\prime}(r).
\end{equation}

\end_inset


\begin_inset Formula 
\begin{equation}
F_{i}=\sqrt{g}\varepsilon_{ijk}J^{j}B^{k}=\sqrt{g}^{-1}\varepsilon_{ijk}\de J^{j}\de B^{k}
\end{equation}

\end_inset

take
\begin_inset Formula 
\begin{equation}
\mathcal{F}_{i}=\sqrt{g}F_{i}=\sqrt{g}\partial_{i}p
\end{equation}

\end_inset

Take
\begin_inset Formula 
\begin{equation}
p^{\prime}(r)=\mathcal{B}^{\ph}(r)(J^{\tht}(\v x)-\iota J^{\ph}(\v x)).
\end{equation}

\end_inset

Thus
\begin_inset Formula 
\begin{equation}
J^{\tht}(\v x)-\iota J^{\ph}(\v x)=f(r)
\end{equation}

\end_inset

must be a flux function.
\end_layout

\begin_layout Section*
Equilibrium
\end_layout

\begin_layout Standard
Unperturbed equilibrium is
\begin_inset Formula 
\begin{align}
\nabla p_{0} & =\v J_{0}\times\v B_{0},\\
\nabla\times\v B_{0} & =\mu_{0}\v J_{0}.
\end{align}

\end_inset

without dependencies on 
\begin_inset Formula $\ph$
\end_inset

.
 Thus
\begin_inset Formula 
\[
\v J_{0}=
\]

\end_inset


\begin_inset Formula 
\begin{equation}
\nabla p_{0}=\v J_{0}\times\v B_{0}
\end{equation}

\end_inset


\end_layout

\begin_layout Section*
Linear perturbation
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\nabla p_{1}=\v J_{1}\times\v B_{0}+\v J_{0}\times\v B_{1}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
p_{mn}^{\prime}\nabla r+imp_{mn}\nabla\tht+inp_{mn}\nabla\ph=\v J_{mn}\times\v B_{0}+\v J_{0}\times\v B_{mn}
\]

\end_inset

Take scalar product with 
\begin_inset Formula $\sqrt{g}\v B_{0}$
\end_inset

 to obtain pressure perturbation
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
imp_{mn}\de B^{\tht}+inp_{mn}\de B^{\ph}=\sqrt{g}\v J_{0}\times\v B_{mn}
\]

\end_inset

or
\begin_inset Formula 
\[
ip_{mn}\de B^{\ph}(m\iota+n)=\sqrt{g}\v J_{0}\times\v B_{mn}
\]

\end_inset


\end_layout

\begin_layout Section*
Cylinder geometry
\end_layout

\begin_layout Standard
Simple model for equilibrium field in the periodic cylinder limit of a torus
 with 
\begin_inset Formula $y=\ph$
\end_inset

.
 Coordinates 
\begin_inset Formula $\v x=(r,\tht,\ph)$
\end_inset

 with
\begin_inset Formula 
\begin{align*}
x(\v x) & =r\cos\tht\\
y(\v x) & =-\ph\\
z(\v x) & =r\sin\tht
\end{align*}

\end_inset

and metric tensor
\begin_inset Formula 
\begin{equation}
g_{ij}=\left(\begin{array}{ccc}
1\\
 & r^{2}\\
 &  & 1
\end{array}\right).
\end{equation}

\end_inset


\end_layout

\end_body
\end_document