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The three-dimensional Variational Moments Equilibrium Code (VMEC) minimizes the energy functional

$ W = \int_{\Omega_p}{ \left ( \frac{1}{2 \mu_0} B^2 + p \right ) dV} $

over the toroidal domain Ωp. The solution is obtained in flux coordinates (s, θ, ζ), related to the cylindrical coordinates (R, φ, Z) by

$ R = \sum{R_{mn}(s) \cos( m\theta - n\zeta)} $
$ Z = \sum{Z_{mn}(s) \sin( m\theta - n\zeta)} $

The code assumes nested flux surfaces. [1] [2]

Uses of the code

Due to its speed in computing the MHD equilibrium problem in 3-D it has become the "de facto" standard code for calculating 3-D equilibria. This means that practically all the laboratories with stellerator devices routinely use it. It has also been used to model tokamak equilibria and lately (2010) it has been applied to reverse field pinches, in particular helical equilibria (non-axisymmetric) in the RFX-Mod. [3]

The code is being used at fusion laboratories all over the world:

  • ORNL, Oak Ridge, TN, USA (code origin)
  • PPPL, Princeton, NJ, USA
  • IPP, at Garching and Greifswald, Germany
  • CRPP, Lausanne, Switzerland
  • NIFS, Toki, Japan
  • RFX, Padova. Italy
  • HSX, Madison, WI, USA
  • LNF, Madrid, Spain

Enhancements / extensions of the code

  • DIAGNO, [4] to calculate the response of magnetic diagnostics
  • MFBE [5]
  • STELLOPT [6]

See also


  1. S.P. Hirschman et al, Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria, Phys. Fluids 26 (1983) 3553
  2. S.P. Hirschman et al, Three-dimensional free boundary calculations using a spectral Green's function method, Computer Physics Communications 43, 1 (1986) 143-155
  3. D. Terranova et al., Self-Organized Helical Equilibria in the RFX-Mod Reversed Field Pinch, Contributions to Plasma Physics 50 (2010) 775–779
  4. H.J. Gardner, Modelling the behaviour of the magnetic field diagnostic coils on the W VII-AS stellarator using a three-dimensional equilibrium code, Nucl. Fusion 30 (1990) 1417
  5. E. Strumberger, Finite-β magnetic field line tracing for Helias configurations, Nucl. Fusion 37 (1997) 19
  6. D.A. Spong et al., Physics issues of compact drift optimized stellarators, Nucl. Fusion 41 (2001) 711