Grad–Shafranov Equation

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.Smith, S. G. L., & Hattori, Y. (2012)
Axisymmetric magnetic vortices with swirl. Communications in Nonlinear Science and Numerical Simulation
17(5), 2101-2107. 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, $F(\psi)=rB_$ and the magnetic field and current are, respectively, given by\begin
\mathbf &= \frac \nabla\psi \times \hat\mathbf_\theta + \frac \hat\mathbf_\theta, \\
\mu_0\mathbf &= \frac \frac \nabla\psi \times \hat\mathbf_\theta - \left frac \left(\frac \frac\right) + \frac \frac\right\hat\mathbf_\theta.
\end

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. $\frac$ produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as
$\mathbf = \left(\frac, -\frac, B_z(x, y)\right),$
or more compactly,
$\mathbf =\nabla A \times \hat + B_z \hat,$
where $A(x,y)\hat$ 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.:
$\nabla p = \mathbf \times \mathbf,$
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 ($\frac = 0$) 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 $\mathbf_\perp \times \mathbf_\perp = 0$, i.e. $\mathbf_\perp$ is parallel to $\mathbf_\perp$.

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

The in plane field is
$\mathbf_\perp = \nabla A \times \hat,$
and using Maxwell–Ampère's equation, the in plane current is given by
$\mathbf_\perp = \frac \nabla B_z \times \hat.$

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

Rearranging the cross products above leads to
$\hat \times \mathbf_\perp = \nabla A - (\mathbf \cdot \nabla A) \mathbf = \nabla A,$
and
$\mathbf_\perp \times B_z\mathbf = \frac(\mathbf\cdot\nabla B_z)\mathbf - \fracB_z\nabla B_z = -\frac B_z\nabla B_z.$

These results can be substituted into the expression for $\nabla p$ to yield:
\nabla p = -\left frac \nabla^2 A\rightnabla A - \frac B_z\nabla B_z.

Since $p$ and $B_z$ are constants along a field line, and functions only of $A$, hence $\nabla p = \frac\nabla A$ and $\nabla B_z = \frac\nabla A$. Thus, factoring out $\nabla A$ and rearranging terms yields the Grad–Shafranov equation:
$\nabla^2 A = -\mu_0 \frac \left(p + \frac\right).$

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 $\vec$ by contravariant basis $(\nabla \Psi, \nabla \phi, \nabla \zeta)$:
$\vec = \nabla\Psi \times \nabla \phi + \bar \nabla\phi,$

we have $\vec$:
$\mu_0 \vec = \nabla \times \vec = -\Delta^* \Psi \nabla \phi+ \nabla\bar \times \nabla \phi \quad \text\ \Delta^* = r\partial_r(r^\partial_r) + \partial^2_\phi \text$

then force balance equation:
$\mu_0 \vec \times \vec= \mu_0 \nabla p\text$

Working out, we have:
$-\Delta^* \Psi = \bar \frac + \mu_0 R^2 \frac \text$

References

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.

{{DEFAULTSORT:Grad-Shafranov equation
Magnetohydrodynamics
Elliptic partial differential equations
Eponymous equations of physics