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 ψ is both a dependent and an independent variable in this equation:
Δ ∗ ψ = − μ 0 R 2 d p d ψ − 1 2 d F 2 d ψ , where μ 0 is the magnetic permeability, p ( ψ ) is the pressure, F ( ψ ) = R B ϕ and the magnetic field and current are, respectively, given by
B → = 1 R ∇ ψ × e ^ ϕ + F R e ^ ϕ , μ 0 J → = 1 R d F d ψ ∇ ψ × e ^ ϕ − 1 R Δ ∗ ψ e ^ ϕ . The elliptic operator Δ ∗ is
Δ ∗ ψ ≡ R 2 ∇ → ⋅ ( 1 R 2 ∇ → ψ ) = R ∂ ∂ R ( 1 R ∂ ψ ∂ R ) + ∂ 2 ψ ∂ 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.
In the following, it is assumed that the system is 2-dimensional with z as the invariant axis, i.e. ∂ ∂ z = 0 for all quantities. Then the magnetic field can be written in cartesian coordinates as
B = ( ∂ A ∂ y , − ∂ A ∂ x , B z ( x , y ) ) , or more compactly,
B = ∇ A × z ^ + B z z ^ , where A ( x , y ) 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 ∇ 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.:
∇ p = j × B , 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 ∇ p is everywhere perpendicular to B). Additionally, the two-dimensional assumption ( ∂ ∂ z = 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 ⊥ × B ⊥ = 0 , i.e. j ⊥ is parallel to B ⊥ .
The right hand side of the previous equation can be considered in two parts:
j × B = j z ( z ^ × B ⊥ ) + j ⊥ × z ^ B z , where the ⊥ 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 = − 1 μ 0 ∇ 2 A . .
The in plane field is
B ⊥ = ∇ A × z ^ ,
and using Maxwell–Ampère's equation, the in plane current is given by
j ⊥ = 1 μ 0 ∇ B z × z ^ .
In order for this vector to be parallel to B ⊥ as required, the vector ∇ B z must be perpendicular to B ⊥ , and B z must therefore, like p , be a field-line invariant.
Rearranging the cross products above leads to
z ^ × B ⊥ = ∇ A − ( z ^ ⋅ ∇ A ) z ^ = ∇ A ,
and
j ⊥ × B z z ^ = B z μ 0 ( z ^ ⋅ ∇ B z ) z ^ − 1 μ 0 B z ∇ B z = − 1 μ 0 B z ∇ B z . These results can be substituted into the expression for ∇ p to yield:
∇ p = − [ 1 μ 0 ∇ 2 A ] ∇ A − 1 μ 0 B z ∇ B z . Since p and B z are constants along a field line, and functions only of A , hence ∇ p = d p d A ∇ A and ∇ B z = d B z d A ∇ A . Thus, factoring out ∇ A and rearranging terms yields the Grad–Shafranov equation:
∇ 2 A = − μ 0 d d A ( p + B z 2 2 μ 0 ) . 