Grad–Shafranov equation explained

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 takes the same form as the Hicks equation from fluid dynamics.^[1] 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). Taking

(r,\theta,z) as the cylindrical coordinates, the flux function \psi is governed by the equation,where \mu _{0} is the magnetic permeability, p(\psi) is the pressure, F(\psi) = rB _{\theta} and the magnetic field and current are, respectively, given by

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 Cartesian coordinates)

In the following, it is assumed that the system is 2-dimensional with

z as the invariant axis, i.e. $\backslash frac$ produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as$$\backslash mathbf\; =\; \backslash left(\backslash frac,\; -\backslash frac,\; B\_z(x,\; y)\backslash right),$$or more compactly,$$\backslash mathbf\; =\backslash nabla\; A\; \backslash times\; \backslash hat\; +\; B\_z\; \backslash hat,$$where A(x,y)\hat{ 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 \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.:$$\backslash nabla\; p\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf,$$where p is the plasma pressure and j is the electric current. It is known that p is a constant along any field line, (again since

\nabla p is everywhere perpendicular to B). Additionally, the two-dimensional assumption ($\backslash frac\; =\; 0$) 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} x B _{\perp} = 0 , i.e. j _{\perp} is parallel to B _{\perp} .

The right hand side of the previous equation can be considered in two parts:$$\backslash mathbf\; \backslash times\; \backslash mathbf\; =\; j\_z\; (\backslash hat\; \backslash times\; \backslash mathbf)\; +\; \backslash mathbf\; \backslash times\; \backslash hatB\_z,$$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$$j\_z\; =\; -\backslash frac\; \backslash nabla^2\; A.$$

The in plane field is$$\backslash mathbf\_\backslash perp\; =\; \backslash nabla\; A\; \backslash times\; \backslash hat,$$and using Maxwell–Ampère's equation, the in plane current is given by$$\backslash mathbf\_\backslash perp\; =\; \backslash frac\; \backslash nabla\; B\_z\; \backslash times\; \backslash hat.$$

In order for this vector to be parallel to

B _{\perp} as required, the vector \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 leads to$$\backslash hat\; \backslash times\; \backslash mathbf\_\backslash perp\; =\; \backslash nabla\; A\; -\; (\backslash mathbf\; \backslash cdot\; \backslash nabla\; A)\; \backslash mathbf\; =\; \backslash nabla\; A,$$and$$\backslash mathbf\_\backslash perp\; \backslash times\; B\_z\backslash mathbf\; =\; \backslash frac(\backslash mathbf\backslash cdot\backslash nabla\; B\_z)\backslash mathbf\; -\; \backslash fracB\_z\backslash nabla\; B\_z\; =\; -\backslash frac\; B\_z\backslash nabla\; B\_z.$$

These results can be substituted into the expression for

\nabla p to yield:$$\backslash nabla\; p\; =\; -\backslash left[\backslash frac\{1\}\{\backslash mu\_0\}\; \backslash nabla^2\; A\backslash right]\backslash nabla\; A\; -\; \backslash frac\; B\_z\backslash nabla\; B\_z.$$

Since

p and B _{z} are constants along a field line, and functions only of A , hence \nabla p = (dp)/(dA) \nabla A and \nabla B _{z} = (d B _{z})/(dA) \nabla A . Thus, factoring out \nabla A and rearranging terms yields the Grad–Shafranov equation:$$\backslash nabla^2\; A\; =\; -\backslash mu\_0\; \backslash frac\; \backslash left(p\; +\; \backslash frac\backslash right).$$

Derivation in contravariant representation

This derivation is only used for Tokamaks, but it can be enlightening. Using the definition of 'The Theory of Toroidally Confined Plasmas 1:3'(Roscoe White), Writing

\vec{B} by contravariant basis (\nabla \Psi, \nabla \phi, \nabla \zeta) :$$\backslash vec\; =\; \backslash nabla\backslash Psi\; \backslash times\; \backslash nabla\; \backslash phi\; +\; \backslash bar\; \backslash nabla\backslash phi,$$

we have

\vec{j} :$$\backslash mu\_0\; \backslash vec\; =\; \backslash nabla\; \backslash times\; \backslash vec=\; -\backslash Delta^*\; \backslash Psi\; \backslash nabla\; \backslash phi+\; \backslash nabla\backslash bar\; \backslash times\; \backslash nabla\; \backslash phi\; \backslash quad\; \backslash text\backslash \; \backslash Delta^*\; =\; r\backslash partial\_r(r^\backslash partial\_r)\; +\; \backslash partial^2\_\backslash phi\; \backslash text$$

then force balance equation:$$\backslash mu\_0\; \backslash vec\; \backslash times\; \backslash vec=\; \backslash mu\_0\; \backslash nabla\; p\backslash text$$

Working out, we have:$$-\backslash Delta^*\; \backslash Psi\; =\; \backslash bar\; \backslash frac\; +\; \backslash mu\_0\; R^2\; \backslash frac\; \backslash text$$

Further reading

• 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
• Haverkort, J.W. (2009) Axisymmetric Ideal MHD Tokamak Equilibria. Notes about the Grad–Shafranov equation, selected aspects of the equation and its analytical solutions.
• Haverkort, J.W. (2009) Axisymmetric Ideal MHD equilibria with Toroidal Flow. Incorporation of toroidal flow, relation to kinetic and two-fluid models, and discussion of specific analytical solutions.

Notes and References

1. Smith, S. G. L., & Hattori, Y. (2012). Axisymmetric magnetic vortices with swirl. Communications in Nonlinear Science and Numerical Simulation, 17(5), 2101-2107.