Struphy equilibrium

ScrewPinch

Straight tokamak (screw pinch) MHD equilibrium for a cylindrical geometry of radius

a

and length

L_{z} = 2 π R_{0}

Governing formula

The profiles in cylindrical coordinates $(r, θ, z)$ with transformation formulae

x &= r\cos\theta\,, y &= r\sin\theta\,, z &= z\,,

are:

\mathbf B(r) &= B_{0}\left( \mathbf e_z + \frac{r}{q(r) R_0}\mathbf e_\theta \right)\,,\qquad q(r) = q_0 + ( q_1 - q_0 )\frac{r^2}{a^2}\,, p(r) &= p0 + \left\{\begin{aligned} &\frac{B_{0}^2 a^2 q_0}{ 2 R_0^2(q_1 - q_0) } \left( \frac{1}{q(r)^2} - \frac{1}{q_1^2} \right) \quad &&\textnormal{if}\quad q_1\neq q_0\neq\infty\,, &\frac{B_{0}^2 a^2}{R_0^2q_0^2} \left(1 - \frac{r^2}{a^2} \right) \quad &&\textnormal{if}\quad q_1= q_0\neq\infty\,, &\beta\frac{B_{0}^2}{2} \quad &&\textnormal{if}\quad q_0= q_1=\infty\,, \end{aligned}\right. n(r) &= n_a + ( 1 - n_a )\left( 1 - \left(\frac{r}{a}\right)^{n_1} \right)^{n_2}\,.

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

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

Pressurex–y · -1.20e-2 to 4.62e-2

Densityx–y · 2.50e-1 to 1.00e+0

Pressurex–z · 1.00e-8 to 4.64e-2

Densityx–z · 2.50e-1 to 9.98e-1

Pressurey–z · 1.00e-8 to 4.64e-2

Densityy–z · 2.50e-1 to 9.98e-1

X-line · pressure / density

Y-line · pressure / density

Z-line · pressure / density

Parameter	Value	Description
`a`	`1`	"Minor" radius (radius of cylinder, default: 1.).
`R0`	`3`	"Major" radius (must be compatible with :math:`L_z=2\pi R_0`, default: 5.).
`B0`	`1`	z-component of magnetic field (constant) (default: 1.).
`q0`	`1.05`	Safety factor at r=0 (use the string "inf" for infinity, default: 1.05).
`q1`	`1.8`	Safety factor at r=a (use the string "inf" for infinity, default: 1.80).
`n1`	`2`	1st shape factor for ion number density profile (default: 0.).
`n2`	`2`	2nd shape factor for ion number density profile (default: 0.).
`na`	`0.25`	Ion nnumber density at r=a (default: 1.).
`p0`	`1e-8`	Pressure offset to avoid numerical issues (default: 1e-8)
`beta`	`0.1`	Plasma beta for :math:`q_0=q_1=\infty` (ratio of kinematic pressure to B^2/2, default: 0.1).