4,383
edits
Flux coordinates: Difference between revisions
no edit summary
No edit summary |
No edit summary |
||
| (21 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
Flux coordinates in the context of magnetic confinement fusion (MCF) is a set of coordinate functions adapted to the shape of the [[Flux surface|flux surfaces]] of the confining magnetic trap. They consist of one flux label, often termed <math>\psi</math> and two angle-like variables <math>\theta, \phi</math> whose constant contours on the flux <math>(\psi({\mathbf x}) = {\textrm constant})</math> surfaces close either poloidaly (<math>\phi</math>) or toroidallly (<math>\theta</math>). | Flux coordinates in the context of magnetic confinement fusion (MCF) is a set of coordinate functions adapted to the shape of the [[Flux surface|flux surfaces]] of the confining magnetic trap. They consist of one flux label, often termed <math>\psi</math> and two angle-like variables <math>\theta, \phi</math> whose constant contours on the flux <math>(\psi({\mathbf x}) = {\textrm constant})</math> surfaces close either poloidaly (<math>\phi</math>) or toroidallly (<math>\theta</math>). | ||
In this coordinates equilibrium vector fields like the magnetic field <math>{\mathbf B}</math> or current density <math>{\mathbf j}</math> have simplified expressions. A particular kind of flux coordinates, generally called [[Flux_coordinates#Magnetic_coordinates|magnetic coordinates]], simplify the <math>{\mathbf B}</math>-field expression further by making field lines look straight in the <math>(\theta, \phi)</math> plane of that family of coordinates.Some popular choices of magnetic coordinate systems are [[Boozer coordinates]] and [[Hamada | In this coordinates, equilibrium vector fields like the magnetic field <math>{\mathbf B}</math> or current density <math>{\mathbf j}</math> have simplified expressions. A particular kind of flux coordinates, generally called [[Flux_coordinates#Magnetic_coordinates|magnetic coordinates]], simplify the <math>{\mathbf B}</math>-field expression further by making field lines look straight in the <math>(\theta, \phi)</math> plane of that family of coordinates. Some popular choices of magnetic coordinate systems are [[Boozer coordinates]] and [[Hamada coordinates]]. | ||
[[Image:CurrentIntegrationCirtuits.png|400px|thumb|right|alt=Flux surface and a <math>\theta</math>-curve and <math>\phi</math>-curve.|Sample flux surface of the [[TJ-II]] [[Stellarator|stellarator]] and a <math>\theta</math>-curve (yellow) and <math>\phi</math>-curve (red).]] | |||
== General curvilinear coordinates == | == General curvilinear coordinates == | ||
| Line 17: | Line 19: | ||
\to | \to | ||
\mathbf{e}_i\cdot\mathbf{e}^j | \mathbf{e}_i\cdot\mathbf{e}^j | ||
= \delta_{i}^{j} | = \delta_{i}^{j}~, | ||
</math> | |||
and therefore relates to the contravariant vectors as | |||
:<math> | |||
\mathbf{e}_i | |||
= \frac{\mathbf{e}^j\times\mathbf{e}^k}{\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k} | = \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 ~, | = \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~, | ||
</math> | </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. | 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. Similarly | ||
:<math> | |||
\mathbf{e}^i = \frac{\mathbf{e}_j\times\mathbf{e}_k}{\sqrt{g}} ~. | |||
</math> | |||
Any vector field <math>\mathbf{B}</math> can be represented as | Any vector field <math>\mathbf{B}</math> can be represented as | ||
| Line 95: | Line 104: | ||
while the curl is | while the curl is | ||
:<math> | :<math> | ||
\nabla\times\mathbf{A} = \frac{\ | \nabla\times\mathbf{A} = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}\mathbf{e}_k \Rightarrow | ||
\left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i} | |||
</math> | </math> | ||
given in terms of the covariant base vectors, where <math>\ | given in terms of the covariant base vectors, where <math>\varepsilon^{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol. | ||
== Flux coordinates == | == Flux coordinates == | ||
| Line 146: | Line 156: | ||
==== Useful properties of FSA ==== | ==== Useful properties of FSA ==== | ||
Some useful properties of the FSA are | Some useful properties of the FSA are | ||
*<math> \langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math> | |||
*<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} = \ | The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggests. This can be seen by simply using Gauss' theorem | ||
*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = | |||
\int_{S(\mathcal{V})}\Gamma\cdot\nabla V \frac{dS}{|\nabla V|} = | |||
\langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.} | |||
</math> | |||
The FSA relates to the conventional volume integral between two surfaces labelled by their volumes <math>V_1</math> and <math>V_2</math> as | |||
*<math> \int_{\mathcal{V}(V_1<V<V_2)} f\; d\mathcal{V} = \int_{V_1}^{V_2} \langle f \rangle\; dV | |||
</math> | |||
whereas the conventional surface integral over a <math>\psi = constant</math> is | |||
*<math> \int_{S(\psi)} f\; dS = \langle f |\nabla V| \rangle | |||
</math> | </math> | ||
Other useful properties are | |||
*<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~. | *<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~. | ||
</math> | </math> | ||
| Line 207: | Line 227: | ||
[[Image:CurrentIntegrationCirtuits.png|256px|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]] | [[Image:CurrentIntegrationCirtuits.png|256px|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]] | ||
[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png|256px|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.]] | [[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png|256px|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 (those enclosing the one drawn here) contribute to the flux of charge through the surface.]] | ||
Note that <math>I</math> is not the current but <math>\mu_0</math> times the current. 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 | Note that <math>I</math> is not the current but <math>\mu_0</math> times the current. 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 | ||
| Line 295: | Line 315: | ||
which can be turned into an algebraic equation on the Fourier components of <math>G</math> | which can be turned into an algebraic equation on the Fourier components of <math>G</math> | ||
:<math> | :<math> | ||
G_{ | G_{mn} = \frac{-i}{\Psi_{pol}'m + \Psi_{tor}'n}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{mn}~. | ||
</math> | </math> | ||
where | where | ||
:<math> | :<math> | ||
G(\psi, \theta_f, \phi_f) = \sum_{n | G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + n\phi_f)} | ||
</math> | </math> | ||
and <math>G_{00} = 0 </math>. | and <math>G_{00} = 0 </math> guarantees periodicity is preserved. | ||
Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are: | Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are: | ||
<ref name='Dhaeseleer'>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</ref> | <ref name='Dhaeseleer'>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}}</ref> | ||
* [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[ | * [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[[doi:10.1063/1.1706651|J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', Phys. Fluids '''5''' (1962) 510]]</ref> In these coordinates, both the magnetic field lines and current lines corresponding to the [[MHD equilibrium]] are straight. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates. | ||
* [[Boozer coordinates]]. <ref>[ | * [[Boozer coordinates]]. <ref>[[doi:10.1063/1.863297|A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]]</ref><ref>[[doi:10.1063/1.863765|A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', Phys. Fluids '''25''' (1982) 520]]</ref> In these coordinates, the magnetic field lines corresponding to the [[MHD equilibrium]] are straight and so are the ''diamagnetic lines '', i.e. the integral lines of <math>\nabla\psi\times\mathbf{B}</math>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates. | ||
== References == | == References == | ||
<references /> | <references /> | ||