Grad-Shafranov equation

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Prerequisites
MHD
 Plasma

The Grad-Shafranov equation (H. Grad and H. Rubin (1958) 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). Interestingly the flux function ψ is both a dependent and an independent variable in this equation:

\Delta^{*}\psi = -\mu_{0}R^{2}\frac{dp}{d\psi}-F\frac{dF}{d\psi}

where μ0 is the magnetic permeability, p(ψ) is the pressure, F(ψ) = RBφ

and the magnetic field and current are given by

\vec{B}=\frac{1}{R}\nabla\psi\times \hat{e_{\phi}}+\frac{F}{R}\hat{e}_{\phi}

\mu_0\vec{J}=\frac{1}{R}\frac{dF}{d\psi}\nabla\psi\times \hat{e_{\phi}}-\frac{1}{R}\Delta^{*}\psi \hat{e}_{\phi}

The elliptic operator

Δ * is given by

\Delta^{*}\psi = 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(ψ) and p(ψ) as well as the boundary conditions.

Derivation:
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

\bold{B} = (\partial A/\partial y,-\partial A /\partial x,B_z(x,y))

or more compactly,

\bold{B} =\nabla A \times \hat{\bold{z}} + \hat{\bold{z}} B_z,

where A(x,y)\hat{\bold{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 ψ mentioned above.)

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

\nabla p = \bold{j} \times \bold{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 \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 \bold{j}_\perp \times \bold{B}_\perp = 0, i.e. \bold{j}_\perp is parallel to \bold{B}_\perp.

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

\bold{j} \times \bold{B} = j_z (\hat{\bold{z}} \times \bold{B_\perp}) +\bold{j_\perp} \times \hat{\bold{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 = -\nabla^2 A/\mu_0.. The in plane field is

\bold{B}_\perp = \nabla A \times \hat{\bold{z}},

and using Ampère's Law the in plane current is given by

\bold{j}_\perp = (1/\mu_0)\nabla B_z \times \hat{\bold{z}}.

In order for this vector to be parallel to \bold{B}_\perp as required, the vector \nabla B_z must be perpendicular to \bold{j}_\perp, and Bz must therefore, like p be a field like invariant.

Rearranging the cross products above, we see that that

\hat{\bold{z}} \times \bold{B}_\perp = \nabla A,

and

\bold{j}_\perp \times B_z\bold{\hat{z}} = -(1/\mu_0)B_z\nabla B_z

These results can be subsituted into the expression for \nabla p to yield:

\nabla p = -[(1/\mu_0) \nabla^2 A]\nabla A-(1/\mu_0)B_z\nabla B_z.

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

\nabla^2 A = -\mu_0 \frac{d}{dA}(p + \frac{B_z^2}{2\mu_0})


[edit] References

  • Grad.H, and Rubin, H. (1958) MHD Equilibrium in an Axisymmetric Toroid. Proceedings of the 2nd UN Conf. on the Peaceful Uses of Atomic Energy, Vol. 31, Vienna: IAEA p.190.
  • Shafranov, V.D. (1966) Plasma equilibrum in a magnetic field, Reviews of Plasma Physics, Vol. 2, New York: Consultants Bureau, p. 103.
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