Struphy equilibrium

AdhocTorusQPsi

Ad hoc tokamak MHD equilibrium with circular concentric flux surfaces.

Governing formula

For a cylindrical coordinate system $(R, ϕ, Z)$ with transformation formulae

x &= R\cos(\phi)\,, &&R = \sqrt{x^2 + y^2}\,, y &= R\sin(\phi)\,, &&\phi = \arctan(y/x)\,, z &= Z\,, &&Z = z\,,

the magnetic field is given by

B = \nabla ψ \times \nabla ϕ + g \nabla ϕ,

where $g = g (R, Z) = - B_{0} R_{0} = co n s t .$ is the toroidal field function, $R_{0}$ the major radius of the torus and $B_{0}$ the on-axis magnetic field. The ad hoc poloidal flux function $ψ = ψ (r)$ is the solution of

\frac{d ψ}{d r} = \frac{B _{0} r}{q ( ψ ( r )) 1 - r ^{2} / R _{0}^{2}}, r = Z^{2} + (R - R_{0})^{2},

for a safety factor profile

q(\psi) &= q_0 + \psi_{\textnormal{norm}}\left[ q_1-q_0+(q_1^\prime-q_1+q_0)\frac{(1-\psi_s)(\psi_{\textnormal{norm}}-1)}{\psi_{\textnormal{norm}}-\psi_s} \right]\,, \psi_{\textnormal{norm}} &= \frac{\psi-\psi(0)}{\psi(a)-\psi(0)}\,, \psi_s &= (q_1^\prime-q_1+q_0)/(q_0^\prime+q_1^\prime-2q_1+2q_0)\,,

where $a$ is the minor radius of the torus.

The pressure and number density profiles are chosen as

p(\psi) &= \frac{\beta B_0^2}{2}\exp\left(-\frac{\psi_{\textnormal{norm}}}{p_1}\right)\,, n(\psi) &= n_a + ( 1 - n_a ) \left( 1 - \psi_{\textnormal{norm}}^{n_1} \right)^{n_2}\,.

Units are those defned in the parameter file (through `struphy.io.options.BaseUnits`).

Coordinate slices

Pressure and density sampled from the generated equilibrium data. Heatmaps use the perceptually uniform Viridis palette.

XY plane at 0

Pressurex–y · 5.49e-2 to 5.49e-2

Densityx–y · 2.00e-1 to 2.00e-1

XZ plane at 0

Pressurex–z · 5.49e-2 to 5.49e-2

Densityx–z · 2.00e-1 to 2.00e-1

YZ plane at 0

Pressurey–z · 5.49e-2 to 5.49e-2

Densityy–z · 2.00e-1 to 2.00e-1

Coordinate-line profiles

X-line · pressure / density

Y-line · pressure / density

Z-line · pressure / density

Parameters

Parameter	Value	Description
`a`	`0.361925`	Minor radius of torus (default: 0.361925).
`R0`	`1`	Major radius of torus (default: 1.).
`B0`	`1`	On-axis (r=0) toroidal magnetic field (default: 1.).
`q0`	`0.35`	Safety factor at r=0 (default: 0.6).
`q1`	`5`	Safety factor at r=a (default: 2.5).
`q0p`	`0.78`	Derivative of safety factor at r=0 (w.r.t. poloidal flux function, default: 0.78).
`q1p`	`5`	Derivative of safety factor at r=a (w.r.t. poloidal flux function, default: 5.00).
`n1`	`2`	1st shape factor for ion number density profile (default: 0.).
`n2`	`1`	2nd shape factor for ion number density profile (default: 0.).
`na`	`0.2`	Ion number density at r=a (default: 1.).
`beta`	`6`	On-axis (r=0) plasma beta (ratio of kinematic pressure to B^2/(2*mu0), default: 0.1).
`p1`	`0.25`	Shape factor for pressure profile, see :meth:`doc_formula` (default: 0.25).
`psi_k`	`3`	Spline degree to be used for interpolation of poloidal flux function (default=3).
`psi_nel`	`50`	Number of cells to be used for interpolation of poloidal flux function (default=50).