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---
title: "Screw pinch"
firstLetter: "S"
publishDate: 2022-03-06
categories:
- Physics
- Plasma physics
date: 2022-01-30T19:27:25+01:00
draft: false
markup: pandoc
---
# Screw pinch
A **pinch** is a type of plasma confinement,
which relies on [magnetic fields](/know/concept/magnetic-field/)
to squeeze the plasma into the desired area.
Examples include tokamaks and stellarators,
although the term *pinch* is typically introduced for simpler 1D confinement.
Suppose that we want to pinch a plasma into a cylindrical shape.
The general way of doing this is called a **screw pinch**.
For simplicity, let the cylinder be infinitely long,
so that it is natural to work in
[cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/)
$(r, \theta, z)$.
Using the framework of ideal [magnetohydrodynamics](/know/concept/magnetohydrodynamics/) (MHD),
let us start by assuming that the fluid is stationary,
and that the confining field $\vb{B}$ is fixed.
From the (ideal) generalized Ohm's law, it then follows
that the [electric field](/know/concept/electric-field/) $\vb{E} = 0$:
$$\begin{aligned}
\vb{u}
= 0
\qquad \qquad
\pdv{\vb{u}}{t}
= 0
\qquad \qquad
\pdv{\vb{B}}{t}
= 0
\qquad \qquad
\vb{E}
= 0
\end{aligned}$$
To get the plasma's equilibrium state for a given $\vb{B}$,
we first solve [Ampère's law](/know/concept/maxwells-equations/)
for the current density $\vb{J}$,
and then the MHD momentum equation for the pressure $p$.
Symmetries should be used whenever possible to reduce these equations:
$$\begin{aligned}
\nabla \cross \vb{B}
= \mu_0 \vb{J}
\qquad \qquad
\vb{J} \cross \vb{B}
= \nabla p
\end{aligned}$$
Note that the latter implies that $\nabla p$ is always orthogonal to $\vb{J}$ and $\vb{B}$,
meaning that the current density and magnetic field must follow
surfaces of constant pressure.
## ϴ-pinch
In a so-called **ϴ-pinch**, the confining field $\vb{B}$
is parallel to the $z$-axis, and its magntiude $B_z$ may only depend on $r$.
Concretely, we have:
$$\begin{aligned}
\vb{B}
= B_z(r) \: \vu{e}_z
\end{aligned}$$
Where $\vu{e}_z$ is the basis vector of the $z$-axis.
This $\vb{B}$ confines the plasma thanks to
the [Lorentz force](/know/concept/lorentz-force/),
which makes charged particles gyrate around magnetic field lines.
Using Ampère's law, we find that the resulting current density $\vb{J}$,
expressed in $(r, \theta, z)$:
$$\begin{aligned}
\vb{J}
= \frac{1}{\mu_0} \nabla \cross \vb{B}
= \frac{1}{\mu_0}
\begin{bmatrix}
\displaystyle \frac{1}{r} \pdv{B_z}{\theta} - \pdv{B_\theta}{z} \\
\displaystyle \pdv{B_r}{z} - \pdv{B_z}{r} \\
\displaystyle \frac{1}{r} \Big( \pdv{(r B_\theta)}{r} - \pdv{B_r}{\theta} \Big)
\end{bmatrix}
= -\frac{1}{\mu_0} \pdv{B_z}{r} \: \vu{e}_\theta
\end{aligned}$$
Where we have used that only $B_z$ is nonzero,
and that it only depends on $r$.
This yields a circular current parallel to $\vu{e}_\theta$,
hence the name *ϴ-pinch*.
Next, we use the MHD momentum equation to find the pressure gradient $\nabla p$.
The cross product is easy to evaluate,
since $\vb{B}$ is parallel to $\vu{e}_z$,
and $\vb{J}$ is parallel to $\vu{e}_\theta$:
$$\begin{aligned}
\nabla p
&= \vb{J} \cross \vb{B}
= J_\theta \vu{e}_\theta \cross B_z \vu{e}_z
= J_\theta B_z \vu{e}_r
= - \frac{1}{\mu_0} \pdv{B_z}{r} B_z \: \vu{e}_r
\end{aligned}$$
Consequently, $\nabla p$ is parallel to $\vu{e}_r$,
and only depends on $r$ through $B_z$.
Along the $r$-direction, the above equation can be rewritten
into the following equilibrium condition:
$$\begin{aligned}
\boxed{
\pdv{r} \bigg( p + \frac{B_z^2}{2 \mu_0} \bigg)
= 0
}
\end{aligned}$$
In other words, the parenthesized expression does not depend on $r$.
## Z-pinch
Meanwhile, in a so-called **Z-pinch**,
we create an $r$-dependent current $\vb{J}$ parallel to the $z$-axis:
$$\begin{aligned}
\vb{J}
= J_z(r) \: \vu{e}_z
\end{aligned}$$
We can then deduce $\vb{B}$ from Ampère's law,
using that only $J_z$ is nonzero,
and that $\pdv*{B_r}{\theta} = 0$ due to circular symmetry:
$$\begin{aligned}
\vb{J}
= \frac{1}{\mu_0} \nabla \cross \vb{B}
= \frac{1}{\mu_0}
\begin{bmatrix}
\displaystyle \frac{1}{r} \pdv{B_z}{\theta} - \pdv{B_\theta}{z} \\
\displaystyle \pdv{B_r}{z} - \pdv{B_z}{r} \\
\displaystyle \frac{1}{r} \Big( \pdv{(r B_\theta)}{r} - \pdv{B_r}{\theta} \Big)
\end{bmatrix}
= \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} \: \vu{e}_z
\end{aligned}$$
Therefore, $\vb{J}$ induces a circular $\vb{B} = B_\theta(r) \: \vu{e}_\theta$,
which confines the plasma for the same reason as in the ϴ-pinch:
the Lorentz force makes particles gyrate around magnetic field lines.
Next, the resulting pressure gradient $\nabla p$ is found from the MHD momentum equation:
$$\begin{aligned}
\nabla p
&= \vb{J} \cross \vb{B}
= J_z \vb{e}_z \cross B_\theta \vb{e}_\theta
= - J_z B_\theta \vu{e}_r
= - \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} B_\theta \: \vu{e}_r
\end{aligned}$$
Once again, $\nabla p$ is parallel to $\vu{e}_r$ and only depends on $r$.
After rearranging, we thus arrive at the following equilibrium condition in the $r$-direction:
$$\begin{aligned}
\boxed{
\pdv{r} \bigg( p + \frac{B_\theta^2}{2 \mu_0} \bigg) + \frac{B_\theta^2}{\mu_0 r}
= 0
}
\end{aligned}$$
## Screw pinch
Thanks to the linearity of electromagnetism,
a ϴ-pinch and Z-pinch can be combined to create a **screw pinch**,
where $\vb{J}$ and $\vb{B}$ both have nonzero $\theta$ and $z$-components.
By performing the above procedure again,
the following equilibrium condition is obtained:
$$\begin{aligned}
\boxed{
\pdv{r} \bigg( p + \frac{B_z^2}{2 \mu_0} + \frac{B_\theta^2}{2 \mu_0} \bigg) + \frac{B_\theta^2}{\mu_0 r}
= 0
}
\end{aligned}$$
Which simply combines the terms of the preceding equations.
Indirectly, this result is relevant for certain types of nuclear fusion reactor,
e.g. the tokamak, which basically consists of a screw pinch bent into a torus.
The resulting equilibrium is given by
the [Grad-Shafranov equation](/know/concept/grad-shafranov-equation/).
## References
1. M. Salewski, A.H. Nielsen,
*Plasma physics: lecture notes*,
2021, unpublished.
|
