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| 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 (physics)|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 [[toroid]]al axisymmetry (the case relevant in a tokamak). The flux function <math>\psi</math> is both a dependent and an [[independent variable]] in this equation:
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| : <math>\Delta^{*}\psi = -\mu_{0}R^{2}\frac{dp}{d\psi}-\frac{1}{2}\frac{dF^2}{d\psi}</math>
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| where <math>\mu_0</math> is the [[magnetic permeability]], <math>p(\psi)</math> is the [[pressure]], <math>F(\psi)=RB_{\phi}</math>
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| and the magnetic field and current are given by | |
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| : <math>\vec{B}=\frac{1}{R}\nabla\psi\times \hat{e}_{\phi}+\frac{F}{R}\hat{e}_{\phi}</math>
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| : <math>\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}</math>
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| The [[elliptic operator]]
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| :<math>\Delta^{*}</math> is given by
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| <math>\Delta^{*}\psi \equiv R^{2} \vec{\nabla} \cdot \left( \frac{1}{R^{2}} \vec{\nabla} \psi \right) = R\frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial \psi}{\partial R}\right)+\frac{\partial^2 \psi}{\partial Z^2}</math>.
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| 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. | |
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| '''Derivation (in slab coordinates):'''<br>
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| To begin we assume that the system is 2-dimensional with z as the invariant axis, i.e. <math>\partial /\partial z = 0</math> for all quantities.
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| Then the magnetic field can be written in cartesian coordinates as
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| :<math> \bold{B} = (\partial A/\partial y,-\partial A /\partial x,B_z(x,y))</math>
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| or more compactly,
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| :<math> \bold{B} =\nabla A \times \hat{\bold{z}} + B_z \hat{\bold{z}}</math>,
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| where <math>A(x,y)\hat{\bold{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.)
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| Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.:
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| :<math>\nabla p = \bold{j} \times \bold{B}</math>,
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| 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 <math>\nabla p</math> is everywhere perpendicular to '''B'''. Additionally, the two-dimensional assumption (<math>\partial / \partial z </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>\bold{j}_\perp \times \bold{B}_\perp = 0</math>, i.e. <math>\bold{j}_\perp</math> is parallel to <math>\bold{B}_\perp</math>.
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| We can break the right hand side of the previous equation into two parts:
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| :<math>\bold{j} \times \bold{B} = j_z (\hat{\bold{z}} \times \bold{B_\perp}) +\bold{j_\perp} \times \hat{\bold{z}}B_z </math>,
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| where the <math>\perp</math> subscript denotes the component in the plane perpendicular to the <math>z</math>-axis. The z component of the current in the above equation can be written in terms of the one dimensional vector potential as
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| <math>j_z = -\nabla^2 A/\mu_0. </math>.
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| The in plane field is <br>
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| :<math>\bold{B}_\perp = \nabla A \times \hat{\bold{z}} </math>,
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| and using Maxwell–Ampère's equation, the in plane current is given by <br>
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| :<math>\bold{j}_\perp = (1/\mu_0)\nabla B_z \times \hat{\bold{z}}</math>. | |
| In order for this vector to be parallel to <math>\bold{B}_\perp</math> as required, the vector <math>\nabla B_z</math> must be perpendicular to <math>\bold{B}_\perp</math>, and <math>B_z</math> must therefore, like <math>p</math> be a field-line invariant.
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| Rearranging the cross products above, we see that <br>
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| :<math>\hat{\bold{z}} \times \bold{B}_\perp = \nabla A -(\bold{\hat z} \cdot\nabla A) \bold{\hat z} = \nabla A</math>,
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| and<br>
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| :<math>\bold{j}_\perp \times B_z\bold{\hat{z}} = -(1/\mu_0)B_z\nabla B_z +(B_z/\mu_0)(\bold{\hat z}\cdot\nabla B_z)\bold{\hat z}=-(1/\mu_0) B_z\nabla B_z.</math>
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| These results can be substituted into the expression for <math>\nabla p </math> to yield: <br>
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| :<math>\nabla p = -\left[(1/\mu_0) \nabla^2 A\right]\nabla A-(1/\mu_0)B_z\nabla B_z. </math>
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| Now, since <math>p</math> and <math>B_z</math> are constants along a field line, and functions only of <math>A</math>, we note that <math>\nabla p = (d p /dA)\nabla A</math> and <math> \nabla B_z = (d B_z/dA)\nabla A</math>. Thus, factoring out <math>\nabla A</math> and rearraging terms we arrive at the '''Grad–Shafranov equation''':<br>
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| : <math>\nabla^2 A = -\mu_0 \frac{d}{dA}\left(p + \frac{B_z^2}{2\mu_0}\right)</math>
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| ==References==
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| * Grad, H., and Rubin, H. (1958) ''[http://www-naweb.iaea.org/napc/physics/2ndgenconf/data/Proceedings%201958/papers%20Vol31/Paper25_Vol31.pdf 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.
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| * Shafranov, V.D. (1966)'' Plasma equilibrium in a magnetic field'', ''Reviews of Plasma Physics'', Vol. 2, New York: Consultants Bureau, p. 103.
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| * Woods, Leslie C. (2004) ''Physics of plasmas'', Weinheim: WILEY-VCH Verlag GmbH & Co. KGaA, chapter 2.5.4
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| {{DEFAULTSORT:Grad-Shafranov equation}}
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| [[Category:Magnetohydrodynamics]]
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| [[Category:Elliptic partial differential equations]]
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