Residual gas analyzer

The Grad–Shafranov equation (H. Grad and H. Rubin (1958); Vitalii Dmitrievich Shafranov (1966)) is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a tokamak. This equation is a two-dimensional, nonlinear, elliptic partial differential equation obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of toroidal axisymmetry (the case relevant in a tokamak). The flux function $\psi$ is both a dependent and an independent variable in this equation:

${\Delta }^{\ast }\psi =-{\mu }_0{R}^2\frac{dp}{d\psi }-\frac{1}{2}\frac{d{F}^2}{d\psi }$

where ${\mu }_0$ is the magnetic permeability, $p(\psi )$ is the pressure, $F(\psi )=R{B}_{\varphi }$

and the magnetic field and current are given by

$\overrightarrow{B}=\frac{1}{R}\mathrm{\nabla }\psi \times {\hat{e}}_{\varphi }+\frac{F}{R}{\hat{e}}_{\varphi }$

${\mu }_0\overrightarrow{J}=\frac{1}{R}\frac{dF}{d\psi }\mathrm{\nabla }\psi \times {\hat{e}}_{\varphi }-\frac{1}{R}{\Delta }^{\ast }\psi {\hat{e}}_{\varphi }$

The elliptic operator

${\Delta }^{\ast }$ is given by

${\Delta }^{\ast }\psi \equiv R^2\overrightarrow{\mathrm{\nabla }}\cdot \left(\frac{1}{R^2}\overrightarrow{\mathrm{\nabla }}\psi \right)=R\frac{\partial }{\partial R}\left(\frac{1}{R}\frac{\partial \psi }{\partial R}\right)+\frac{{\partial }^2\psi }{\partial Z^2}$.

The nature of the equilibrium, whether it be a tokamak, reversed field pinch, etc. is largely determined by the choices of the two functions $F(\psi )$ and $p(\psi )$ as well as the boundary conditions.

Derivation (in slab coordinates):
To begin we assume that the system is 2-dimensional with z as the invariant axis, i.e. $\partial /\partial z=0$ for all quantities. Then the magnetic field can be written in cartesian coordinates as

$𝐁=(\partial A/\partial y,-\partial A/\partial x,B_z(x,y))$

or more compactly,

$𝐁=\mathrm{\nabla }A\times \widehat{𝐳}+B_z\widehat{𝐳}$,

where $A(x,y)\widehat{𝐳}$ is the vector potential for the in-plane (x and y components) magnetic field. Note that based on this form for B we can see that A is constant along any given magnetic field line, since $\mathrm{\nabla }A$ is everywhere perpendicular to B. (Also note that -A is the flux function $\psi$ mentioned above.)

Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.:

$\mathrm{\nabla }p=𝐣\times 𝐁$,

where p is the plasma pressure and j is the electric current. Note from the form of this equation that we also know p is a constant along any field line, (again since $\mathrm{\nabla }p$ is everywhere perpendicular to B. Additionally, the two-dimensional assumption ($\partial /\partial z$) means that the z- component of the left hand side must be zero, so the z-component of the magnetic force on the right hand side must also be zero. This means that ${𝐣}_{\perp }\times {𝐁}_{\perp }=0$, i.e. ${𝐣}_{\perp }$ is parallel to ${𝐁}_{\perp }$.

We can break the right hand side of the previous equation into two parts:

$𝐣\times 𝐁=j_z(\widehat{𝐳}\times {𝐁}_{\mathbf{\perp }})+{𝐣}_{\mathbf{\perp }}\times \widehat{𝐳}{B}_z$,

where the $\perp$ subscript denotes the component in the plane perpendicular to the $z$-axis. The z component of the current in the above equation can be written in terms of the one dimensional vector potential as $j_z=-{\mathrm{\nabla }}^{2}A/{\mu }_0.$. The in plane field is

${𝐁}_{\perp }=\mathrm{\nabla }A\times \widehat{𝐳}$,

and using Maxwell–Ampère's equation, the in plane current is given by

${𝐣}_{\perp }=(1/{\mu }_0)\mathrm{\nabla }{B}_z\times \widehat{𝐳}$.

In order for this vector to be parallel to ${𝐁}_{\perp }$ as required, the vector $\mathrm{\nabla }{B}_z$ must be perpendicular to ${𝐁}_{\perp }$, and $B_z$ must therefore, like $p$ be a field-line invariant.

Rearranging the cross products above, we see that

$\widehat{𝐳}\times {𝐁}_{\perp }=\mathrm{\nabla }A-(\widehat{𝐳}\cdot \mathrm{\nabla }A)\widehat{𝐳}=\mathrm{\nabla }A$,

and

${𝐣}_{\perp }\times B_z\widehat{𝐳}=-(1/{\mu }_0)B_z\mathrm{\nabla }{B}_z+(B_z/{\mu }_0)(\widehat{𝐳}\cdot \mathrm{\nabla }{B}_z)\widehat{𝐳}=-(1/{\mu }_0)B_z\mathrm{\nabla }{B}_z.$

These results can be substituted into the expression for $\mathrm{\nabla }p$ to yield:

$\mathrm{\nabla }p=-[(1/{\mu }_0){\mathrm{\nabla }}^{2}A]\mathrm{\nabla }A-(1/{\mu }_0)B_z\mathrm{\nabla }{B}_z.$

Now, since $p$ and $B_z$ are constants along a field line, and functions only of $A$, we note that $\mathrm{\nabla }p=(dp/dA)\mathrm{\nabla }A$ and $\mathrm{\nabla }{B}_z=(d{B}_z/dA)\mathrm{\nabla }A$. Thus, factoring out $\mathrm{\nabla }A$ and rearraging terms we arrive at the Grad–Shafranov equation:

${\mathrm{\nabla }}^{2}A=-{\mu }_0\frac{d}{dA}(p+\frac{B_z^2}{2{\mu }_0})$

References

• Grad, H., and Rubin, H. (1958) Hydromagnetic Equilibria and Force-Free Fields. Proceedings of the 2nd UN Conf. on the Peaceful Uses of Atomic Energy, Vol. 31, Geneva: IAEA p. 190.
• Shafranov, V.D. (1966) Plasma equilibrium in a magnetic field, Reviews of Plasma Physics, Vol. 2, New York: Consultants Bureau, p. 103.
• Woods, Leslie C. (2004) Physics of plasmas, Weinheim: WILEY-VCH Verlag GmbH & Co. KGaA, chapter 2.5.4