Английская Википедия: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.[1] 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 <math>(r,\theta,z)</math> as the cylindrical coordinates, the flux function <math>\psi</math> is governed by the equation,Шаблон:Equation box 1where <math>\mu_0</math> is the magnetic permeability, <math>p(\psi)</math> is the pressure, <math>F(\psi)=rB_{\theta}</math> and the magnetic field and current are, respectively, given by<math display="block">\begin{align}
\mathbf{B} &= \frac{1}{r} \nabla\psi \times \hat\mathbf{e}_\theta + \frac{F}{r} \hat\mathbf{e}_\theta, \\
\mu_0\mathbf{J} &= \frac{1}{r} \frac{dF}{d\psi} \nabla\psi \times \hat\mathbf{e}_\theta - \left[\frac{\partial}{\partial r} \left(\frac{1}{r} \frac{\partial \psi}{\partial r}\right) + \frac{1}{r} \frac{\partial^2 \psi}{\partial z^2}\right] \hat\mathbf{e}_\theta.
\end{align}</math>
The nature of the equilibrium, whether it be a tokamak, reversed field pinch, etc. is largely determined by the choices of the two functions <math>F(\psi)</math> and <math>p(\psi)</math> as well as the boundary conditions.
Derivation (in Cartesian coordinates)
In the following, it is assumed that the system is 2-dimensional with <math>z</math> as the invariant axis, i.e. <math display="inline">\frac{\partial}{\partial z}</math> produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as <math display="block"> \mathbf{B} = \left(\frac{\partial A}{\partial y}, -\frac{\partial A}{\partial x}, B_z(x, y)\right),</math> or more compactly, <math display="block"> \mathbf{B} =\nabla A \times \hat{\mathbf{z}} + B_z \hat{\mathbf{z}},</math> where <math>A(x,y)\hat{\mathbf{z}}</math> 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 <math>\nabla A</math> is everywhere perpendicular to B. (Also note that -A is the flux function <math>\psi</math> mentioned above.)
Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.: <math display="block">\nabla p = \mathbf{j} \times \mathbf{B},</math> 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 <math>\nabla p</math> is everywhere perpendicular to B). Additionally, the two-dimensional assumption (<math display="inline">\frac{\partial}{\partial z} = 0</math>) 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 <math>\mathbf{j}_\perp \times \mathbf{B}_\perp = 0</math>, i.e. <math>\mathbf{j}_\perp</math> is parallel to <math>\mathbf{B}_\perp</math>.
The right hand side of the previous equation can be considered in two parts: <math display="block">\mathbf{j} \times \mathbf{B} = j_z (\hat{\mathbf{z}} \times \mathbf{B_\perp}) + \mathbf{j_\perp} \times \hat{\mathbf{z}}B_z ,</math> where the <math>\perp</math> subscript denotes the component in the plane perpendicular to the <math>z</math>-axis. The <math>z</math> component of the current in the above equation can be written in terms of the one-dimensional vector potential as <math display="block">j_z = -\frac{1}{\mu_0} \nabla^2 A. </math>
The in plane field is <math display="block">\mathbf{B}_\perp = \nabla A \times \hat{\mathbf{z}}, </math> and using Maxwell–Ampère's equation, the in plane current is given by <math display="block">\mathbf{j}_\perp = \frac{1}{\mu_0} \nabla B_z \times \hat{\mathbf{z}}.</math>
In order for this vector to be parallel to <math>\mathbf{B}_\perp</math> as required, the vector <math>\nabla B_z</math> must be perpendicular to <math>\mathbf{B}_\perp</math>, and <math>B_z</math> must therefore, like <math>p</math>, be a field-line invariant.
Rearranging the cross products above leads to <math display="block">\hat{\mathbf{z}} \times \mathbf{B}_\perp = \nabla A - (\mathbf{\hat z} \cdot \nabla A) \mathbf{\hat z} = \nabla A,</math> and <math display="block">\mathbf{j}_\perp \times B_z\mathbf{\hat{z}} = \frac{B_z}{\mu_0}(\mathbf{\hat z}\cdot\nabla B_z)\mathbf{\hat z} - \frac{1}{\mu_0}B_z\nabla B_z = -\frac{1}{\mu_0} B_z\nabla B_z.</math>
These results can be substituted into the expression for <math>\nabla p</math> to yield: <math display="block">\nabla p = -\left[\frac{1}{\mu_0} \nabla^2 A\right]\nabla A - \frac{1}{\mu_0} B_z\nabla B_z.</math>
Since <math>p</math> and <math>B_z</math> are constants along a field line, and functions only of <math>A</math>, hence <math>\nabla p = \frac{dp}{dA}\nabla A</math> and <math> \nabla B_z = \frac{d B_z}{dA}\nabla A</math>. Thus, factoring out <math>\nabla A</math> and rearranging terms yields the Grad–Shafranov equation: <math display="block">\nabla^2 A = -\mu_0 \frac{d}{dA} \left(p + \frac{B_z^2}{2\mu_0}\right).</math>
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.
- ↑ Smith, S. G. L., & Hattori, Y. (2012). Axisymmetric magnetic vortices with swirl. Communications in Nonlinear Science and Numerical Simulation, 17(5), 2101-2107.
