Struphy equilibrium

AdhocTorus

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 ( r ) 1 - r ^{2} / R _{0}^{2}}, r = Z^{2} + (R - R_{0})^{2},

for some given safety factor profile. Two profiles in terms of the on-axis $q_{0} \equiv q (r = 0)$ and edge $q_{1} \equiv q (r = a)$ safety factor values are available ( $a$ is the minor radius of the torus):

q(r) &= \left\{\begin{aligned} &q_0 + ( q_1 - q_0 )\frac{r^2}{a^2} \quad &&\textnormal{if} \quad q_\textnormal{kind}=0\,, &\frac{q_0}{1-\left(1-\frac{r^2}{a^2}\right)^{\frac{q_1}{q_0}}}\frac{r^2}{a^2} \quad &&\textnormal{if} \quad q_\textnormal{kind}=1\,. &q_0 + l\frac{r}{a} ( q_1 - q_0 )\frac{r^2}{a^2} \quad &&\textnormal{if} \quad q_\textnormal{kind}=2\,, \end{aligned}\right.

The pressure profile

p^\prime(r) &= -\frac{B_0^2}{R_0^2}\frac{r\left[2q(r)-rq^\prime(r)\right]}{q(r)^3} \quad &&\textnormal{if} \quad p_\textnormal{kind}=0\,, p(r) &= \beta \frac{B_{0}^2}{2} \left( p_0 - p_1 \frac{r^2}{a^2} - p_2 \frac{r^4}{a^4} \right) \quad &&\textnormal{if} \quad p_\textnormal{kind}=1\,,

is either the exact solution of the MHD equilibrium condition in the cylindrical limit ( $p_{kind} = 0$ ) or an monotonically decreasing adhoc profile for some given on-axis plasma beta ( $p_{kind} = 1$ ). Finally, the number density profile is chosen as

n (r) = n_{a} + (1 - n_{a}) (1 - (\frac{r}{a})^{n_{1}})^{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 · -2.64e+0 to 4.16e-2

Densityx–y · -5.93e+0 to -1.01e+0

XZ plane at 0

Pressurex–z · -3.40e+0 to -3.59e-1

Densityx–z · -6.81e+0 to -2.20e+0

YZ plane at 0

Pressurey–z · -3.40e+0 to -3.59e-1

Densityy–z · -6.81e+0 to -2.20e+0

Coordinate-line profiles

X-line · pressure / density

Y-line · pressure / density

Z-line · pressure / density

Parameters

Parameter	Value	Description
`a`	`1`	Minor radius of torus (default: 1.).
`R0`	`3`	Major radius of torus (default: 3.).
`B0`	`2`	On-axis (r=0) toroidal magnetic field (default: 2.).
`q_kind`	`0`	Which safety factor profile, see :meth:`doc_formula` (0, 1 or 2, default: 0).
`q0`	`1.71`	Safety factor at r=0 (default: 1.71).
`q1`	`1.87`	Safety factor at r=a (default: 1.87).
`l`	`0`	Linear term factor for q profile if q_kind=2 (default: 0.).
`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.).
`p_kind`	`1`	Kind of pressure profile, see :meth:`doc_formula` (0 or 1, default: 1).
`p0`	`1`	constant factor for ad hoc pressure profile (default: 1.).
`p1`	`0.1`	1st shape factor for ad hoc pressure profile (default: 0.).
`p2`	`0.1`	2nd shape factor for ad hoc pressure profile (default: 0.).
`beta`	`0.179`	On-axis (r=0) plasma beta if p_kind=1 (ratio of kinematic pressure to B^2/(2*mu0), default: 0.179).
`psi_k`	`3`	Spline degree to be used for interpolation of poloidal flux function (if q_kind=1, default=3).
`psi_nel`	`50`	Number of cells to be used for interpolation of poloidal flux function (if q_kind=1, default=50).