The Grad–Shafranov equation (
H. Grad and H. Rubin (1958);
Vitalii Dmitrievich Shafranov (1966)) is the equilibrium equation in ideal
magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
(MHD) for a two dimensional
plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral), a green translucent silica mineral
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood pla ...
, for example the axisymmetric toroidal plasma in a
tokamak
A tokamak (; russian: токамáк; otk, 𐱃𐰸𐰢𐰴, Toḳamaḳ) is a device which uses a powerful magnetic field to confine plasma in the shape of a torus. The tokamak is one of several types of magnetic confinement devices being d ...
. This equation takes the same form as the
Hicks equation In fluid dynamics, Hicks equation, sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation, is a partial differential equation that describes the distribution of stream function for axisymmetric inviscid fluid, named after Wi ...
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
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
,
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
,
elliptic partial differential equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
wher ...
obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of
toroid
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
al axisymmetry (the case relevant in a tokamak). Taking
as the cylindrical coordinates, the flux function
is governed by the equation,
where
is the
magnetic permeability
In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
,
is the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
and the magnetic field and current are, respectively, given by
The nature of the equilibrium, whether it be a
tokamak
A tokamak (; russian: токамáк; otk, 𐱃𐰸𐰢𐰴, Toḳamaḳ) is a device which uses a powerful magnetic field to confine plasma in the shape of a torus. The tokamak is one of several types of magnetic confinement devices being d ...
, reversed field pinch, etc. is largely determined by the choices of the two functions
and
as well as the boundary conditions.
Derivation (in Cartesian coordinates)
In the following, it is assumed that the system is 2-dimensional with
as the invariant axis, i.e.
produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as
or more compactly,
where
is the
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
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
is everywhere perpendicular to B. (Also note that -A is the flux function
mentioned above.)
Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.:
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
is everywhere perpendicular to B). Additionally, the two-dimensional assumption (
) 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
, i.e.
is parallel to
.
The right hand side of the previous equation can be considered in two parts:
where the
subscript denotes the component in the plane perpendicular to the
-axis. The
component of the current in the above equation can be written in terms of the one-dimensional vector potential as
The in plane field is
and using Maxwell–Ampère's equation, the in plane current is given by
In order for this vector to be parallel to
as required, the vector
must be perpendicular to
, and
must therefore, like
, be a field-line invariant.
Rearranging the cross products above leads to
and
These results can be substituted into the expression for
to yield:
Since
and
are constants along a field line, and functions only of
, hence
and
. Thus, factoring out
and rearranging terms yields the Grad–Shafranov equation:
References
* 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