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
)
.
