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<ref>F. Alladio, F. Chrisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', Nucl. Fusion '''26''' (1986) 1143</ref>
<ref>F. Alladio, F. Chrisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', Nucl. Fusion '''26''' (1986) 1143</ref>
<ref>[http://dx.doi.org/10.1016/0010-4655(94)90112-0 B.Ph. van Milligen and A. Lopez Fraguas, ''Expansion of vacuum magnetic fields in toroidal harmonics'', Computer Physics Communications '''81''', Issues 1-2 (1994) 74-90]</ref>
<ref>[http://dx.doi.org/10.1016/0010-4655(94)90112-0 B.Ph. van Milligen and A. Lopez Fraguas, ''Expansion of vacuum magnetic fields in toroidal harmonics'', Computer Physics Communications '''81''', Issues 1-2 (1994) 74-90]</ref>
== General curvilinear coordinates ==
Here we briefly review the basic definitions of a general [[:Wikipedia:Curvilinear coordinates | curvilinear coordinate system]] for later convenience when discussing toroidal flux coordinates and magnetic coordinates.
=== Function coordinates and basis vector ===
Given the spatial dependence of a coordinate set <math>(\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))</math>
we can calculate the contravariant basis vectors
:<math>
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}
</math>
and the dual covariant basis defined as
:<math>
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}}
\to
\mathbf{e}_i\cdot\mathbf{e}^j
= \delta_{i}^{j} \to \mathbf{e}_i
= \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|}
= \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~,
</math>
where <math>(i,j,k)</math> are cyclic permutations of <math>(1,2,3)</math> and we have used the notation <math>(u^1, u^2, u^3) = (\psi,\theta,\phi)</math>. The Jacobian <math>\sqrt{g}</math> is defined below.
Any vector field <math>\mathbf{B}</math> can be represented as
:<math>
\mathbf{B}
= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i
= B^i\mathbf{e}_i
</math>
or
:<math>
\mathbf{B}
= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i
= B_i\mathbf{e}^i ~.
</math>
In particular any basis vector <math>\mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j</math>. The metric tensor is defined as
:<math>
g_{ij}
= \mathbf{e}_i\cdot\mathbf{e}_j
\; ; \;
g^{ij}
= \mathbf{e}^i\cdot\mathbf{e}^j
\; ; \;
g^j_i 
= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~.
</math>
=== Jacobian ===
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as
:<math>
J = \det\left(\frac{\partial(x,y,z)}{\partial(\psi,\theta,\phi)}\right) = \frac{\partial\mathbf{x}}{\partial{\psi}}\cdot\frac{\partial\mathbf{x}}{\partial{\theta}} \times \frac{\partial\mathbf{x}}{\partial{\phi}}
</math>
and that of the inverse transformation
:<math>
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi}
</math>
It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>
== Flux coordinates ==
A flux coordinate set is one that includes a [[Flux surface|flux surface]] label as a coordinate. A flux surface label is a function that is constant and single valued on each flux surface. In our naming of the general curvilinear coordinates we have already adopted the usual flux coordinate convention for toroidal equilibrium with nested flux surfaces with <math>\psi</math> being the flux surface label and <math>\theta, \phi</math> are <math>2\pi</math>-periodic poloidal and toroidal-like angles.
Different flux surface labels can be chosen like toroidal <math>(\Psi_{tor})</math> or poloidal <math>(\Psi_{pol})</math> magnetic fluxes or the volume contained within the flux surface <math>V</math>. By single valued we mean to ensure that any flux label <math>\psi_1 = f(\psi_2)</math> is a monotonous function of any other flux label  <math>\psi_2</math>, so that the function  <math>f</math> is invertible at least in a volume containing the region of interest. We will denote a generic flux surface label by <math>\psi</math>.
=== Flux Surface Average ===
The flux surface average of a function <math>\Phi</math> is defined as the limit
:<math>
\langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V}
</math>
where <math>\delta \mathcal{V}</math> is the volume confined between two flux surfaces. It is therefore a ''volume average'' over an infinitesimal spatial region rather than a surface average. To avoid confusion, we denote volume elements or domains with the calligraphic <math>\mathcal{V}</math>. Capital <math>V</math> is reserved for the flux label (coordinate) defined as the volume within a flux surface.
Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math>
:<math>
\langle\Phi\rangle
= \lim_{\delta \mathcal{V} \to 0} \frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; \sqrt{g} d\psi d\theta d\phi
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi
</math>
or, noting that <math>\langle 1\rangle = 1</math>, we have <math>\frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi</math> and
we get to a more practical form of the Flux Surface Average
:<math>
\langle\Phi\rangle
= \frac{\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi}
{\int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi}
</math>
Note that <math>dS = |\nabla\psi|\sqrt{g}d\theta d\phi</math>, so the FSA is a surface integral ''weighted by'' <math>|\nabla V|^{-1}</math> :
:<math>
\langle\Phi\rangle
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi
= \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; dS
= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS
</math>
Applying Gauss' theorem to the definition of FSA we get to the identity
:<math>
\langle\nabla\cdot\Gamma\rangle
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V}
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{S(\delta \mathcal{V})} \Gamma\cdot \frac{\nabla V}{|\nabla V|}dS
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\left(\langle\Gamma\cdot\nabla V\rangle_{S(V+\delta \mathcal{V})} - \langle\Gamma\cdot\nabla V\rangle_{S(V)} \right)
= \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~.
</math>
==== Useful properties of FSA ====
Some useful properties of the FSA are
*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle  = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math>
*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} =  \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle</math>
*<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'}
</math>
*<math> \langle \mathbf{B}\cdot\nabla f \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla V\cdot\mathbf{B} = 0 </math>
*<math> \langle \mathbf{B}\cdot\nabla \theta\rangle =2\pi\frac{d\Psi_{pol}}{dV} \qquad (\mathrm{Note:}~ \theta(\mathbf{x})~\mathrm{is~not~single~valued})
</math>
*<math> \langle \mathbf{B}\cdot\nabla \phi\rangle =2\pi\frac{d\Psi_{tor}}{dV} \qquad (\mathrm{Note:}~ \phi(\mathbf{x})~\mathrm{is~not~single~valued})
</math>
where <math>V' = \frac{dV}{d\psi}</math>.
=== Magnetic field representation in flux coordinates ===
==== Contravariant From ====
Any [[:Wikipedia: solenoidal vector field| solenoidal vector field]] <math>\mathbf{B}</math>  can be written as
<math> \mathbf{B} = \nabla\alpha\times\nabla\nu </math>
called its Clebsch representation. For a magnetic field with flux surfaces <math>(\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)</math> we can choose, say, <math>\alpha</math> to be the flux surface label <math>\psi</math>
:<math>
\mathbf{B} = \nabla\psi\times\nabla\nu
</math>
Field lines are then given as the intersection of the constant-<math>\psi</math> and constant-<math>\nu</math> surfaces. This form provides a general expression for <math>\mathbf{B}</math> in terms of the covariant basis vectors of a flux coordinate system
:<math>
\mathbf{B} = \frac{\partial\nu}{\partial\theta}\nabla\psi\times\nabla\theta + \frac{\partial\nu}{\partial\phi}\nabla\psi\times\nabla\phi =  \frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\theta}\mathbf{e}_\phi -\frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\phi}\mathbf{e}_\theta = B^\phi\mathbf{e}_\phi + B^\theta\mathbf{e}_\theta~.
</math>
in terms of the function <math>\nu</math>, sometimes referred to as the magnetic field's ''stream function''.
It is worthwhile to note that the Clebsch form of <math> \mathbf{B} </math> corresponds to a [[:Wikipedia: Magnetic potential|magnetic vector potential]]
<math> \mathbf{A} = \nu\nabla\psi </math> (or <math> \mathbf{A} = \psi\nabla\nu </math> as they differ only by the Gauge transformation <math> \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)</math>).
The general form of the stream function is
:<math>
\nu(\psi,\theta,\phi)
= \frac{1}{2\pi}(\Psi_{tor}'\theta
- \Psi_{pol}'\phi)
+ \tilde{\nu}(\psi,\theta,\phi)
</math>
where <math>\tilde{\nu}</math> is a differentiable function periodic in the two angles. This general form can be derived by using the fact that  <math> \mathbf{B}</math> is a physical function (hence singe-valued). The specific form for the coefficients in front of the secular terms (i.e. the non-periodic terms) can be obtained from the [[Toroidal coordinates#Useful properties of FSA|FSA properties ]].
==== Covariant Form ====
If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, where <math> \mathbf{j}</math> is the current density , then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as
:<math>
\mathbf{B} = \beta\nabla\psi + \nabla\chi
</math>
where <math>\chi</math> is identified as the magnetic ''scalar'' potential. Its general form is
:<math>
\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi)
</math>
[[Image:CurrentIntegrationCirtuits.png|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]]
[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png|thumb|right|alt=Sample surface for the definition of the current though a disc.|Sample surface for the definition of the current though a disc. Note that only the current of more external surfaces contribute to the flux of charge through the surface.]]
The functional dependence on the angular variables is again motivated by the single-valuedness of the magnetic field. The particular form of the coefficients can be obtained noting that
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}
= \int_{\partial S}\mathbf{B}\cdot d\mathbf{l}
= \oint(\beta\nabla\psi + \nabla\chi)\cdot d\mathbf{l}
= \oint(\beta d\psi + d\chi)
</math>
and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}
= \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~.
</math>
If we now chose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get
:<math>
I_{pol}^d = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 0, \Delta\phi = 2\pi)~.
</math>
here the superscript <math>d</math> is meant to indicate the flux is computed through a disc limited by the integration line, as opposed to the ribbon limited by the integration line on one side and the magnetic axis on the other that was used for the definition of poloidal magnetic flux <math>\Psi_{pol}</math> above these lines. 
Similarly
:<math>
I_{tor} = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 2\pi, \Delta\phi = 0)~.
</math>
===== Contravariant Form of the current density =====
Taking the curl of the covariant form of <math>\mathbf{B}</math> the equilibrium current density <math>\mathbf{j}</math> can be written as
: <math>
\mathbf{j} = \nabla\psi\times\nabla\eta~.
</math>
By very similar arguments as those used for <math>\mathbf{B}</math> (note that both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for <math>\eta</math> is
:<math>
\eta(\psi,\theta,\phi) = \frac{1}{2\pi}({I}_{tor}'\theta
- {I}_{pol}'\phi)
+ \tilde{\eta}(\psi,\theta,\phi)~.
</math>
Note that the poloidal current is now defined through a ribbon and not a disc.


== Magnetic coordinates ==
== Magnetic coordinates ==
Magnetic coordinates are a particular type of flux coordinates in which the magnetic field lines are straight lines. In mathematical terms this implies that the periodic part of the magnetic field's stream function is zero in these coordinates so the magnetic field reads
Magnetic coordinates are a particular type of [[flux coordinates]] in which the magnetic field lines are straight lines. Magnetic coordinates adapt to the magnetic field, and therefore to the [[MHD equilibrium]] (also see [[Flux surface]]).  
:<math>
\mathbf{B} =  \nabla\psi\times \left( \frac{\Psi_{tor}'}{2\pi}\theta_f
- \frac{\Psi_{pol}'}{2\pi}\phi_f \right)~.
</math>
Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system
:<math>
B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}}
</math>
are not flux functions, but their quotient is
:<math>
\frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~,
</math>
<math>\iota</math> being the [[rotational transform]].
 
Any transformation of the angles of the from
:<math>
\theta_F = \theta_f +\Psi_{pol}' G(\psi, \theta_f, \phi_f)\; ;\quad \phi_F = \phi_f +\Psi_{tor}' G(\psi, \theta_f, \phi_f)
</math>
 
where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''.
 
 
 
Magnetic coordinates adapt to the magnetic field, and therefore to the [[MHD equilibrium]] (also see [[Flux surface]]).  
Magnetic coordinates simplify the description of the magnetic field.  
Magnetic coordinates simplify the description of the magnetic field.  
In 3 dimensions (not assuming axisymmetry), the most commonly used coordinate systems are:
In 3 dimensions (not assuming axisymmetry), the most commonly used coordinate systems are:

Revision as of 10:24, 19 August 2010

A simple toroidal coordinate system

Coordinate systems used in toroidal systems:

Cartesian coordinates

(X, Y, Z) [1]

Cylindrical coordinates

(R, φ, Z), where [2]

  • R2 = X2 + Y2, and
  • tan φ = Y/X.

φ is called the toroidal angle (and not the cylindrical angle, at least not in the context of magnetic confinement).

Cylindrical symmetry (symmetry under rotation over φ) is referred to as axisymmetry.

Simple toroidal coordinates

(r, φ, θ), where

  • R = R0 + r cos θ, and
  • Z = r sin θ

R0 corresponds to the torus axis and is called the major radius, while r is called the minor radius, and θ the poloidal angle.

In order to adapt this simple system better to the magnetic surfaces of an axisymmetric MHD equilibrium, it may be enhanced by [3]

  • letting R0 depend on r (to account for the Shafranov shift of flux surfaces) [4]
  • adding ellipticity (ε), triangularity (κ), etc. (to account for non-circular flux surface cross sections)

Toroidal coordinates

(ζ, η, φ), where [5] [6]

R=Rpsinh⁡ζcosh⁡ζ−cos⁡η
Z=Rpsin⁡ηcosh⁡ζ−cos⁡η

where Rp is the pole of the coordinate system. Surfaces of constant ζ are tori with major radii R = Rp/tanh ζ and minor radii r = Rp/sinh ζ. At R = Rp, ζ = ∞, while at infinity and at R = 0, ζ = 0. The coordinate η is a poloidal angle and runs from 0 to 2π. This system is orthogonal.

The Laplace equation separates in this system of coordinates, thus allowing an expansion of the vacuum magnetic field in toroidal harmonics. [7] [8]

Magnetic coordinates

Magnetic coordinates are a particular type of flux coordinates in which the magnetic field lines are straight lines. Magnetic coordinates adapt to the magnetic field, and therefore to the MHD equilibrium (also see Flux surface). Magnetic coordinates simplify the description of the magnetic field. In 3 dimensions (not assuming axisymmetry), the most commonly used coordinate systems are: [9]

  • Hamada coordinates. [10][11] In these coordinates, both the field lines and current lines corresponding to the MHD equilibrium are straight. Referring to the definitions above, both ν~ and η~ are zero in Hamada coordinates.
  • Boozer coordinates. [12][13] In these coordinates, the field lines corresponding to the MHD equilibrium are straight and so are the diamagnetic lines , i.e. the integral lines of ∇ψ×𝐁. Referring to the definitions above, both ν~ and χ~ are zero in Boozer coordinates.

These two coordinate systems are related. [14]

References

  1. ↑ Wikipedia:Cartesian coordinate system
  2. ↑ Wikipedia:Cylindrical coordinate system
  3. ↑ R.L. Miller et al, Noncircular, finite aspect ratio, local equilibrium model, Phys. Plasmas 5 (1998) 973
  4. ↑ R.D. Hazeltine, J.D. Meiss, Plasma confinement, Courier Dover Publications (2003) ISBN 0486432424
  5. ↑ Morse and Feshbach, Methods of theoretical physics, McGraw-Hill, New York, 1953 ISBN 007043316X
  6. ↑ Wikipedia:Toroidal coordinates
  7. ↑ F. Alladio, F. Chrisanti, Analysis of MHD equilibria by toroidal multipolar expansions, Nucl. Fusion 26 (1986) 1143
  8. ↑ B.Ph. van Milligen and A. Lopez Fraguas, Expansion of vacuum magnetic fields in toroidal harmonics, Computer Physics Communications 81, Issues 1-2 (1994) 74-90
  9. ↑ W.D. D'haeseleer, Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory, Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193
  10. ↑ S. Hamada, Nucl. Fusion 2 (1962) 23
  11. ↑ J.M. Greene and J.L Johnson, Stability Criterion for Arbitrary Hydromagnetic Equilibria, Phys. Fluids 5 (1962) 510
  12. ↑ A.H. Boozer, Plasma equilibrium with rational magnetic surfaces, Phys. Fluids 24 (1981) 1999
  13. ↑ A.H. Boozer, Establishment of magnetic coordinates for a given magnetic field, Phys. Fluids 25 (1982) 520
  14. ↑ K. Miyamoto, Controlled fusion and plasma physics, Vol. 21 of Series in Plasma Physics, CRC Press (2007) ISBN 1584887095
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