Flux coordinates

General curvilinear coordinates

Here we briefly review the basic definitions of a general 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   we can calculate the contravariant basis vectors

 

and the dual covariant basis defined as

 

where   are cyclic permutations of   and we have used the notation  . The Jacobian   is defined below.

Any vector field   can be represented as

 

or

 

In particular any basis vector  . The metric tensor is defined as

 

The metric tensors can be used to raise or lower indices. Take

 

so that

 

Jacobian

The Jacobian of the coordinate transformation   is defined as

 

and that of the inverse transformation

 

It can be seen that [1]  

Gradient, Divergence and Curl in curvilinear coordinates

The gradient of a funcion f is naturally given in the contravariant base vectors:

 

The divergence of a vector \mathbf{A} is best expressed in terms of its contravariant components

 

while the curl is

 

given in temr of the covariant base vectors, where   is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol.

Flux coordinates

A flux coordinate set is one that includes a 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   being the flux surface label and   are  -periodic poloidal and toroidal-like angles.

Different flux surface labels can be chosen like toroidal   or poloidal   magnetic fluxes or the volume contained within the flux surface  . By single valued we mean to ensure that any flux label   is a monotonous function of any other flux label  , so that the function   is invertible at least in a volume containing the region of interest. We will denote a generic flux surface label by  .

To avoid ambiguity in the sign of line and surface integrals we impose  , the toroidal angle increases in the clockwise direction when seen from above and the poloidal angle increases such that  .

Flux Surface Average

The flux surface average of a function   is defined as the limit

 

where   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  . Capital   is reserved for the flux label (coordinate) defined as the volume within a flux surface.

Introducing the differential volume element  

 

or, noting that  , we have   and we get to a more practical form of the Flux Surface Average

 

Note that  , so the FSA is a surface integral weighted by   :

 

Applying Gauss' theorem to the definition of FSA we get to the identity

 

Useful properties of FSA

Some useful properties of the FSA are

  •  
  •  
  •  
  •  
  •  
  •  
  •  

In the above  .

Magnetic field representation in flux coordinates

Contravariant From

Any solenoidal vector field   can be written as   called its Clebsch representation. For a magnetic field with flux surfaces   we can choose, say,   to be the flux surface label  

 

Field lines are then given as the intersection of the constant-  and constant-  surfaces. This form provides a general expression for   in terms of the covariant basis vectors of a flux coordinate system

 

in terms of the function  , sometimes referred to as the magnetic field's stream function.

It is worthwhile to note that the Clebsch form of   corresponds to a magnetic vector potential   (or   as they differ only by the Gauge transformation  ).

The general form of the stream function is

 

where   is a differentiable function periodic in the two angles. This general form can be derived by using the fact that   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 FSA properties .

Covariant Form

If we consider an equilibrium magnetic field such that  , where   is the current density , then both   and   and the magnetic field can be written as

 

where   is identified as the magnetic scalar potential. Its general form is

 
 
Sample integration circuits for the current definitions.
 
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

 

and choosing an integration circuit contained within a flux surface  . Then we get

 

If we now chose a toroidal circuit   we get

 

here the superscript   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   above these lines. Similarly

 
Contravariant Form of the current density

Taking the curl of the covariant form of   the equilibrium current density   can be written as

 

By very similar arguments as those used for   (note that both   and   are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for   is

 

Note that the poloidal current is now defined through a ribbon and not a disc. The two currents are related as   implies

 

where the integral is performed along the magnetic axis.

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

 

Now a field line is given by   and  .

Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system

 

are not flux functions, but their quotient is

 

  being the rotational transform. In a magnetic coordinate system the poloidal   and toroidal   components of the magnetic field are individually divergence-less.

Transforming between Magnetic coordinates systems

There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from

 

where   is periodic in the angles, preserves the straightness of the field lines (as can be easily checked by direct substitution). The spatial function  , is called the generating function. It can be obtained from a magnetic differential equation if we know the Jacobians of the two magnetic coordinate systems   and  . In fact taking   on any of the transformation of the angles and using the known expressions for the contravariant components of   in magnetic coordinates we get

 

Particular choices of G can be that simplify the description of other fields. The most commonly used magnetic coordinate systems are: [1]

  • Hamada coordinates. [2][3] In these coordinates, both the field lines and current lines corresponding to the MHD equilibrium are straight. Referring to the definitions above, both   and   are zero in Hamada coordinates.
  • Boozer coordinates. [4][5] 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  . Referring to the definitions above, both   and   are zero in Boozer coordinates.

References

  1. 1.0 1.1 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
  2. S. Hamada, Nucl. Fusion 2 (1962) 23
  3. J.M. Greene and J.L Johnson, Stability Criterion for Arbitrary Hydromagnetic Equilibria, Phys. Fluids 5 (1962) 510
  4. A.H. Boozer, Plasma equilibrium with rational magnetic surfaces, Phys. Fluids 24 (1981) 1999
  5. A.H. Boozer, Establishment of magnetic coordinates for a given magnetic field, Phys. Fluids 25 (1982) 520