--- 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.