---
title: "Screw pinch"
date: 2022-03-06
categories:
- Physics
- Plasma physics
layout: "concept"
---

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 $\ipdv{B_r}{\theta} = 0$ 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.