Ideal MHD

These fields describe the quantities used to compute the ideal MHD equilibria, or quantities derived from the equilibria. For quantities describing fast-ion confinement, see Fast-ion confinement.

File layout

Each HDF5 file holds a set of stellarator configurations. Every field is a top-level dataset whose first axis runs over configurations, so row i of every dataset belongs to the same configuration. The uuid field identifies a configuration across all files. Field tables list types and shapes per configuration.

All quantities are in SI units (m, T, Wb, Pa, A, s, m/s), and angles are in radians unless noted. Every dataset also carries a short description and its units as HDF5 attributes, e.g. f["modB"].attrs["units"].

Grids and array shapes

Gridded fields use 15 radial points (s1d), 41 toroidal angles over one field period (phi1d), and 41 poloidal angles (theta1d). They are stored flattened in C order:

modB = f["modB"][i].reshape(15, 41, 41)             # [s, phi, theta]
boundary = f["boundary_xyz"][i].reshape(41, 41, 3)   # [phi, theta, xyz]
axis = f["xyz_axis"][i].reshape(41, 3)              # [phi, xyz]

Identifiers

HDF5 name	Symbol	Type	Description
uuid	–	str	Unique identifier of the configuration. It is the same in every file.
source_dataset	–	str	Source dataset the configuration came from, e.g. quasr, constellaration. See Sources.
source_id	–	str	Identifier of the configuration in its source dataset.
timestamp	–	str	When the particle tracing was run.

Geometry

boundary_full_res_fourier_modes describes the boundary as a Fourier series, stellarator-symmetric and in SIMSOPT SurfaceRZFourier order:

\[ R(\theta, \phi) = \sum_{m,n} R_{mn} \cos(m\theta - n\,n_\text{fp}\,\phi), \qquad Z(\theta, \phi) = \sum_{m,n} Z_{mn} \sin(m\theta - n\,n_\text{fp}\,\phi). \]

The vector is \([R_{mn}]\) followed by \([Z_{mn}]\). For \(R_{mn}\), \(m = 0\) runs over \(n = 0, \dots, 20\), and each \(m = 1, \dots, 20\) runs over \(n = -20, \dots, 20\), giving 841 terms. \(Z_{mn}\) has the same layout without \(Z_{00}\), giving 840 terms.

HDF5 name	Symbol	Type	Description
boundary_full_res_fourier_modes	\(R_{mn},\ Z_{mn}\)	array (1681,)	Fourier coefficients of the plasma boundary [m], zero-padded to mpol = ntor = 20, as described above.
boundary_full_res_mpol, boundary_full_res_ntor	\(M,\ N\)	int	Resolution of boundary_full_res_fourier_modes (always 20).
boundary_mpol, boundary_ntor	–	int	Resolution of the boundary in its original representation.
boundary_xyz	\(\mathbf{x}(\phi, \theta)\)	array (5043,)	Cartesian points on the plasma boundary over one field period [m]. Reshape to (41, 41, 3), indexed \((\phi, \theta, xyz)\).
boundary_quadpoints_phi	\(\phi\)	array (41,)	Toroidal angles of boundary_xyz, in turns, \([0, 1/n_\text{fp})\).
boundary_quadpoints_theta	\(\theta\)	array (41,)	Poloidal angles of boundary_xyz, in turns, \([0, 1)\).
nfp	\(n_\text{fp}\)	int	Number of field periods.
stellsym	–	bool	Whether the configuration is stellarator-symmetric.
major_radius	\(R_0\)	float	Major radius of the plasma boundary [m].
minor_radius	\(a\)	float	Effective minor radius of the plasma boundary [m].
aspect_ratio	\(A = R_0 / a\)	float	Aspect ratio.
volume	\(V\)	float	Plasma volume [m³].
xyz_axis	\(\mathbf{r}_\text{axis}(\phi)\)	array (123,)	Cartesian points on the magnetic axis over one field period [m], at the angles boundary_quadpoints_phi. Reshape to (41, 3).
axis_helicity	–	int	Helicity of the magnetic axis.
integrated_axis_torsion	\(\int \tau \, d\ell\)	float	Torsion of the magnetic axis, integrated along its length over one field period.

Magnetic field

HDF5 name	Symbol	Type	Description
s1d	\(s\)	array (15,)	Normalized toroidal flux of the radial grid: 15 linearly spaced values from \(10^{-4}\) to 1 (\(s = 1\) is the boundary).
theta1d	\(\theta\)	array (41,)	Poloidal angle grid, \([0, 2\pi)\).
phi1d	\(\phi\)	array (41,)	Toroidal angle grid over one field period, \([0, 2\pi/n_\text{fp})\).
modB	\(\lvert B \rvert(s, \phi, \theta)\)	array (25215,)	Magnetic field strength on the grid, in VMEC angles [T]. Reshape to (15, 41, 41).
modB_boozer	\(\lvert B \rvert(s, \zeta_B, \theta_B)\)	array (25215,)	Magnetic field strength on the grid, in Boozer angles [T]. Reshape to (15, 41, 41).
R_boozer	\(R(s, \zeta_B, \theta_B)\)	array (25215,)	Cylindrical radius of the Boozer grid points [m]. Reshape to (15, 41, 41).
Z_boozer	\(Z(s, \zeta_B, \theta_B)\)	array (25215,)	Cylindrical height of the Boozer grid points [m]. Reshape to (15, 41, 41).
nu_boozer	\(\nu(s, \zeta_B, \theta_B)\)	array (25215,)	Difference \(\nu = \zeta_B - \phi\) between the Boozer and cylindrical toroidal angles [rad], so the cylindrical angle of a point is \(\phi = \zeta_B - \nu\). Reshape to (15, 41, 41).
max_modB	\(B_\text{max}\)	float	Maximum field strength [T].
min_modB	\(B_\text{min}\)	float	Minimum field strength [T].
mirror_ratio	\(B_\text{max} / B_\text{min}\)	float	Mirror ratio.
volume_avg_modB	\(\langle B \rangle\)	float	Volume-averaged field strength [T].
min_L_grad_B	\(\min L_{\nabla B}\)	float	Minimum magnetic gradient scale length on the boundary [m].
toroidal_flux_edge	\(\Phi_\text{edge}\)	float	Toroidal flux enclosed by the boundary [Wb].

Equilibrium profiles

Profiles are sampled on the 15 radial points of s1d.

HDF5 name	Symbol	Type	Description
enforce_vacuum	–	bool	True for vacuum equilibria (\(\beta = 0\)). Their particles are traced with the vacuum guiding-center equations (\(I = K = 0\)).
plasma_beta	\(\langle \beta \rangle\)	float	Volume-averaged plasma beta.
pressure	\(p(s)\)	array (15,)	Pressure profile [Pa].
d_pressure_d_s	\(dp/ds\)	array (15,)	Radial derivative of the pressure [Pa].
iota	\(\iota(s)\)	array (15,)	Rotational transform.
toroidal_current	\(I_\text{tor}(s)\)	array (15,)	Enclosed toroidal current profile [A].
jdotb_vmec	\(\langle \mathbf{J} \cdot \mathbf{B} \rangle\)	array (15,)	Flux-surface-averaged parallel current from VMEC [T·A/m²].
boozer_G	\(G(s)\)	array (15,)	Boozer covariant toroidal field component [T·m].
boozer_I	\(I(s)\)	array (15,)	Boozer covariant poloidal field component [T·m].
vacuum_well	\(W\)	float	Magnetic well depth. Positive values mean a magnetic well, which is favorable.
mercier_criterion	\(D_\text{Merc}(s)\)	array (15,)	Mercier stability criterion. Positive values mean stable.

Quasi-symmetry

HDF5 name	Symbol	Type	Description
sqrt_boozer_qs_error	\(\sqrt{f_\text{QS}}(s)\)	array (15,)	Quasi-symmetry error: the root-sum-square of the Boozer \(\lvert B \rvert\) modes that break quasi-symmetry with helicity (1, axis_helicity), divided by that of the symmetric modes.

VMEC settings

Solver settings used to compute each equilibrium.

HDF5 name	Symbol	Type	Description
vmec_mpol, vmec_ntor	–	int	Poloidal and toroidal Fourier resolution.
vmec_ntheta, vmec_nzeta	–	int	Real-space poloidal and toroidal grid sizes.
vmec_ns_array	–	array (10,)	Radial resolution at each multigrid stage.
vmec_niter_array	–	array (10,)	Maximum iterations at each stage.
vmec_ftol_array	–	array (10,)	Force-residual tolerance at each stage.
vmec_delt	–	float	Solver time step.
vmec_ncurr	–	int	0 = prescribed \(\iota\) profile, 1 = prescribed current profile.
vmec_signgs	–	int	Sign of the Jacobian.