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diff --git a/source/know/concept/screw-pinch/index.md b/source/know/concept/screw-pinch/index.md index c7477b3..0a787dc 100644 --- a/source/know/concept/screw-pinch/index.md +++ b/source/know/concept/screw-pinch/index.md @@ -19,13 +19,13 @@ 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/) -$(r, \theta, z)$. +$$(r, \theta, z)$$. 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. +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/) $\vb{E} = 0$: +that the [electric field](/know/concept/electric-field/) $$\vb{E} = 0$$: $$\begin{aligned} \vb{u} @@ -41,10 +41,10 @@ $$\begin{aligned} = 0 \end{aligned}$$ -To get the plasma's equilibrium state for a given $\vb{B}$, +To get the plasma's equilibrium state for a given $$\vb{B}$$, we first solve [Ampère's law](/know/concept/maxwells-equations/) -for the current density $\vb{J}$, -and then the MHD momentum equation for the pressure $p$. +for the current density $$\vb{J}$$, +and then the MHD momentum equation for the pressure $$p$$. Symmetries should be used whenever possible to reduce these equations: $$\begin{aligned} @@ -55,15 +55,15 @@ $$\begin{aligned} = \nabla p \end{aligned}$$ -Note that the latter implies that $\nabla p$ is always orthogonal to $\vb{J}$ and $\vb{B}$, +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 $r$. +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: $$\begin{aligned} @@ -71,13 +71,13 @@ $$\begin{aligned} = B_z(r) \: \vu{e}_z \end{aligned}$$ -Where $\vu{e}_z$ is the basis vector of the $z$-axis. -This $\vb{B}$ confines the plasma thanks to +Where $$\vu{e}_z$$ is the basis vector of the $$z$$-axis. +This $$\vb{B}$$ confines the plasma thanks to the [Lorentz force](/know/concept/lorentz-force/), which makes charged particles gyrate around magnetic field lines. -Using Ampère's law, we find that the resulting current density $\vb{J}$, -expressed in $(r, \theta, z)$: +Using Ampère's law, we find that the resulting current density $$\vb{J}$$, +expressed in $$(r, \theta, z)$$: $$\begin{aligned} \vb{J} @@ -91,15 +91,15 @@ $$\begin{aligned} = -\frac{1}{\mu_0} \pdv{B_z}{r} \: \vu{e}_\theta \end{aligned}$$ -Where we have used that only $B_z$ is nonzero, -and that it only depends on $r$. -This yields a circular current parallel to $\vu{e}_\theta$, +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 $\nabla p$. +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 $\vu{e}_\theta$: +since $$\vb{B}$$ is parallel to $$\vu{e}_z$$, +and $$\vb{J}$$ is parallel to $$\vu{e}_\theta$$: $$\begin{aligned} \nabla p @@ -109,9 +109,9 @@ $$\begin{aligned} = - \frac{1}{\mu_0} \pdv{B_z}{r} B_z \: \vu{e}_r \end{aligned}$$ -Consequently, $\nabla p$ is parallel to $\vu{e}_r$, -and only depends on $r$ through $B_z$. -Along the $r$-direction, the above equation can be rewritten +Consequently, $$\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} @@ -121,22 +121,22 @@ $$\begin{aligned} } \end{aligned}$$ -In other words, the parenthesized expression does not depend on $r$. +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: +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: +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: $$\begin{aligned} \vb{J} @@ -150,11 +150,11 @@ $$\begin{aligned} = \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} \: \vu{e}_z \end{aligned}$$ -Therefore, $\vb{J}$ induces a circular $\vb{B} = B_\theta(r) \: \vu{e}_\theta$, +Therefore, $$\vb{J}$$ induces a circular $$\vb{B} = B_\theta(r) \: \vu{e}_\theta$$, which confines the plasma for the same reason as in the ϴ-pinch: the Lorentz force makes particles gyrate around magnetic field lines. -Next, the resulting pressure gradient $\nabla p$ is found from the MHD momentum equation: +Next, the resulting pressure gradient $$\nabla p$$ is found from the MHD momentum equation: $$\begin{aligned} \nabla p @@ -164,8 +164,8 @@ $$\begin{aligned} = - \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} B_\theta \: \vu{e}_r \end{aligned}$$ -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: +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: $$\begin{aligned} \boxed{ @@ -179,7 +179,7 @@ $$\begin{aligned} 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. +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: |