Categories: Physics, Plasma physics.

Grad-Shafranov equation

Nuclear fusion reactors tend to have a torus shape, in which the plasma is confined by a pinch, i.e. by magnetic fields chosen so that the Lorentz force stops particles escaping. Effectively, we are taking a cylindrical screw pinch and bending it into a torus.

We would like to find the equilibrium state of the plasma in the general case of a reactor with toroidal symmetry. Using ideal magnetohydrodynamics (MHD), we start by assuming that the fluid is stationary, and that the confining field \(\vb{B}\) is fixed:

\[\begin{aligned} \vb{u} = 0 \qquad \qquad \pdv{\vb{u}}{t} = 0 \qquad \qquad \pdv{\vb{B}}{t} = 0 \qquad \qquad \vb{E} = 0 \end{aligned}\]

Notice that \(\vb{E} = 0\) is a result of the ideal generalized Ohm’s law. Under these assumptions, the relevant MHD equations to be solved are Gauss’ law for magnetism, Ampère’s law, and the MHD momentum equation, respectively:

\[\begin{aligned} 0 = \nabla \cdot \vb{B} \qquad \qquad \mu_0 \vb{J} = \nabla \cross \vb{B} \qquad \qquad \nabla p = \vb{J} \cross \vb{B} \end{aligned}\]

The goal is to analyze them in this order, exploiting toroidal symmetry along the way, to arrive at a general equilibrium condition. Cylindrical polar coordinates \((r, \theta, z)\) are a natural choice, with the \(z\)-axis running through the middle of the torus.

As preparation, it is a good idea to write \(\vb{B}\) as the curl of a magnetic vector potential \(\vb{A}\), which looks like this in cylindrical polar coordinates:

\[\begin{aligned} \vb{B} = \nabla \cross \vb{A} = \begin{bmatrix} \displaystyle \frac{1}{r} \pdv{A_z}{\theta} - \pdv{A_\theta}{z} \\ \displaystyle \pdv{A_r}{z} - \pdv{A_z}{r} \\ \displaystyle \frac{1}{r} \Big( \pdv{(r A_\theta)}{r} - \pdv{A_r}{\theta} \Big) \end{bmatrix} = \begin{bmatrix} \displaystyle - \pdv{A_\theta}{z} \\ \displaystyle \pdv{A_r}{z} - \pdv{A_z}{r} \\ \displaystyle \frac{1}{r} \pdv{(r A_\theta)}{r} \end{bmatrix} \end{aligned}\]

Here, it is convenient to define the so-called stream function \(\psi\) as follows:

\[\begin{aligned} \boxed{ \psi \equiv r A_\theta } \end{aligned}\]

Such that \(\vb{B}\) can be written as below, where we will regard \(B_\theta\) as a given quantity:

\[\begin{aligned} \vb{B} = \begin{bmatrix} \displaystyle -\frac{1}{r} \pdv{\psi}{z} \\ B_\theta \\ \displaystyle \frac{1}{r} \pdv{\psi}{r} \end{bmatrix} \qquad \mathrm{where} \qquad B_\theta = \pdv{A_r}{z} - \pdv{A_z}{r} \end{aligned}\]

Inserting this into Gauss’ law, we see that it is trivially satisfied, thanks to circular symmetry guaranteeing that \(\pdv*{B_\theta}{\theta} = 0\):

\[\begin{aligned} 0 = \nabla \cdot \vb{B} &= - \frac{1}{r} \pdv{r} \bigg( \frac{r}{r} \pdv{\psi}{z} \bigg) + \frac{1}{r} \pdv{B_\theta}{\theta} + \pdv{z} \bigg( \frac{1}{r} \pdv{\psi}{r} \bigg) \\ &= - \frac{1}{r} \pdv{\psi}{r}{z} + \frac{1}{r} \pdv{\psi}{z}{r} = 0 \end{aligned}\]

What matters is that we have expressions for the components of \(\vb{B}\). Moving on, to find the current density \(\vb{J}\), we use Ampère’s law and symmetry to get:

\[\begin{aligned} \vb{J} = \frac{1}{\mu_0} \nabla \cross \vb{B} = \frac{1}{\mu_0} \begin{bmatrix} \displaystyle \frac{1}{r} \pdv{B_z}{\theta} - \pdv{B_\theta}{z} \\ \displaystyle \pdv{B_r}{z} - \pdv{B_z}{r} \\ \displaystyle \frac{1}{r} \Big( \pdv{(r B_\theta)}{r} - \pdv{B_r}{\theta} \Big) \end{bmatrix} = \frac{1}{\mu_0} \begin{bmatrix} \displaystyle 0 \\ \displaystyle \pdv{B_r}{z} - \pdv{B_z}{r} \\ \displaystyle \frac{1}{r} \pdv{(r B_\theta)}{r} \end{bmatrix} \end{aligned}\]

Where we have assumed that \(B_\theta\) depends only on \(r\), not \(z\) or \(\theta\). Substituting this into the MHD momentum equation gives the following pressure gradient \(\nabla p\):

\[\begin{aligned} \nabla p &= \vb{J} \cross \vb{B} = \begin{bmatrix} J_\theta B_z - J_z B_\theta \\ J_z B_r - J_r B_z \\ J_r B_\theta - J_\theta B_r \end{bmatrix} = \begin{bmatrix} J_\theta B_z - J_z B_\theta \\ J_z B_r \\ - J_\theta B_r \end{bmatrix} \end{aligned}\]

Now, the idea is to focus on this \(r\)-component to get an equation for \(\psi\), whose solution can then be used to calculate the \(\theta\) and \(z\)-components of \(\nabla p\). Therefore, we evaluate:

\[\begin{aligned} \pdv{p}{r} &= J_\theta B_z - J_z B_\theta \\ &= \frac{1}{\mu_0} \bigg( \pdv{B_r}{z} - \pdv{B_z}{r} \bigg) B_z - \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} B_\theta \\ &= - \frac{1}{\mu_0} \bigg( \pdv{z}\Big(\frac{1}{r} \pdv{\psi}{z}\Big) + \pdv{r}\Big(\frac{1}{r} \pdv{\psi}{r}\Big) \bigg) \frac{1}{r} \pdv{\psi}{r} - \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} B_\theta \\ &= - \frac{1}{\mu_0 r} \bigg( \frac{1}{r} \pdv[2]{\psi}{z} + \pdv{r} \Big( \frac{1}{r} \pdv{\psi}{r} \Big) \bigg) \pdv{\psi}{r} - \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} B_\theta \end{aligned}\]

By using the chain rule to rewrite \(\pdv*{r} = (\pdv*{\psi}{r}) \; \pdv*{\psi}\), we get \(\pdv*{\psi}{r}\) in each term:

\[\begin{aligned} \pdv{\psi}{r} \pdv{p}{\psi} &= - \frac{1}{\mu_0 r} \bigg( \frac{1}{r} \pdv[2]{\psi}{z} + \pdv{r} \Big( \frac{1}{r} \pdv{\psi}{r} \Big) \bigg) \pdv{\psi}{r} - \frac{1}{\mu_0 r} \pdv{\psi}{r} \pdv{(r B_\theta)}{\psi} B_\theta \end{aligned}\]

Dividing out \(\pdv*{\psi}{r}\) and multiplying by \(\mu_0 r^2\) leads us to the Grad-Shafranov equation, which gives the equilibrium condition of a plasma in a toroidal reactor:

\[\begin{aligned} \boxed{ \pdv[2]{\psi}{z} + r \pdv{r} \bigg( \frac{1}{r} \pdv{\psi}{r} \bigg) = - \mu_0 r^2 \pdv{p}{\psi} - r \pdv{(r B_\theta)}{\psi} B_\theta } \end{aligned}\]

Weirdly, \(\psi\) appears both as an unknown and as a differentiation variable, but this equation can still be solved analytically by assuming a certain \(\psi\)-dependence of \(p\) and \(r B_\theta\).

Suppose that \(B_\theta\) is induced by a poloidal electrical current \(I_\mathrm{pol}\), i.e. a current around the “tube” of the torus, then, assuming \(I_\mathrm{pol}\) only depends on \(r\), we have:

\[\begin{aligned} B_\theta = \frac{\mu_0 I_\mathrm{pol}(r)}{2 \pi r} \end{aligned}\]

Inserting this into the Grad-Shafranov equation yields its following alternative form:

\[\begin{aligned} \boxed{ \pdv[2]{\psi}{z} + r \pdv{r} \bigg( \frac{1}{r} \pdv{\psi}{r} \bigg) = - \mu_0 r^2 \pdv{p}{\psi} - \frac{\mu_0^2}{8 \pi^2} \pdv{I_\mathrm{pol}^2}{\psi} } \end{aligned}\]

References

  1. M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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