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Grad–Shafranov equation

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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, for example the axisymmetric toroidal plasma in a tokamak. This equation takes the same form as the Hicks equation from fluid dynamics. 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 ( r , θ , z ) {\displaystyle (r,\theta ,z)} as the cylindrical coordinates, the flux function ψ {\displaystyle \psi } is governed by the equation,

∂ 2 ψ ∂ r 2 − 1 r ∂ ψ ∂ r + ∂ 2 ψ ∂ z 2 = − μ 0 r 2 d p d ψ − 1 2 d F 2 d ψ , {\displaystyle {\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=-\mu _{0}r^{2}{\frac {dp}{d\psi }}-{\frac {1}{2}}{\frac {dF^{2}}{d\psi }},}

where μ 0 {\displaystyle \mu _{0}} is the magnetic permeability, p ( ψ ) {\displaystyle p(\psi )} is the pressure, F ( ψ ) = r B θ {\displaystyle F(\psi )=rB_{\theta }} and the magnetic field and current are, respectively, given by B = 1 r ∇ ψ × e ^ θ + F r e ^ θ , μ 0 J = 1 r d F d ψ ∇ ψ × e ^ θ − [ ∂ ∂ r ( 1 r ∂ ψ ∂ r ) + 1 r ∂ 2 ψ ∂ z 2 ] e ^ θ . {\displaystyle {\begin{aligned}\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{\hat {\mathbf {e} }}_{\theta }.\end{aligned}}}

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 ) {\displaystyle F(\psi )} and p ( ψ ) {\displaystyle p(\psi )} as well as the boundary conditions.

Derivation (in Cartesian coordinates)

In the following, it is assumed that the system is 2-dimensional with z {\displaystyle z} as the invariant axis, i.e. ∂ ∂ z {\textstyle {\frac {\partial }{\partial z}}} produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as B = ( ∂ A ∂ y , − ∂ A ∂ x , B z ( x , y ) ) , {\displaystyle \mathbf {B} =\left({\frac {\partial A}{\partial y}},-{\frac {\partial A}{\partial x}},B_{z}(x,y)\right),} or more compactly, B = ∇ A × z ^ + B z z ^ , {\displaystyle \mathbf {B} =\nabla A\times {\hat {\mathbf {z} }}+B_{z}{\hat {\mathbf {z} }},} where A ( x , y ) z ^ {\displaystyle A(x,y){\hat {\mathbf {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 {\displaystyle \nabla A} is everywhere perpendicular to B. (Also note that -A is the flux function ψ {\displaystyle \psi } mentioned above.)

Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.: ∇ p = j × B , {\displaystyle \nabla p=\mathbf {j} \times \mathbf {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 {\displaystyle \nabla p} is everywhere perpendicular to B). Additionally, the two-dimensional assumption ( ∂ ∂ z = 0 {\textstyle {\frac {\partial }{\partial 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 {\displaystyle \mathbf {j} _{\perp }\times \mathbf {B} _{\perp }=0} , i.e. j ⊥ {\displaystyle \mathbf {j} _{\perp }} is parallel to B ⊥ {\displaystyle \mathbf {B} _{\perp }} .

The right hand side of the previous equation can be considered in two parts: j × B = j z ( z ^ × B ⊥ ) + j ⊥ × z ^ B z , {\displaystyle \mathbf {j} \times \mathbf {B} =j_{z}({\hat {\mathbf {z} }}\times \mathbf {B_{\perp }} )+\mathbf {j_{\perp }} \times {\hat {\mathbf {z} }}B_{z},} where the ⊥ {\displaystyle \perp } subscript denotes the component in the plane perpendicular to the z {\displaystyle z} -axis. The z {\displaystyle 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 . {\displaystyle j_{z}=-{\frac {1}{\mu _{0}}}\nabla ^{2}A.}

The in plane field is B ⊥ = ∇ A × z ^ , {\displaystyle \mathbf {B} _{\perp }=\nabla A\times {\hat {\mathbf {z} }},} and using Maxwell–Ampère's equation, the in plane current is given by j ⊥ = 1 μ 0 ∇ B z × z ^ . {\displaystyle \mathbf {j} _{\perp }={\frac {1}{\mu _{0}}}\nabla B_{z}\times {\hat {\mathbf {z} }}.}

In order for this vector to be parallel to B ⊥ {\displaystyle \mathbf {B} _{\perp }} as required, the vector ∇ B z {\displaystyle \nabla B_{z}} must be perpendicular to B ⊥ {\displaystyle \mathbf {B} _{\perp }} , and B z {\displaystyle B_{z}} must therefore, like p {\displaystyle p} , be a field-line invariant.

Rearranging the cross products above leads to z ^ × B ⊥ = ∇ A − ( z ^ ⋅ ∇ A ) z ^ = ∇ A , {\displaystyle {\hat {\mathbf {z} }}\times \mathbf {B} _{\perp }=\nabla A-(\mathbf {\hat {z}} \cdot \nabla A)\mathbf {\hat {z}} =\nabla 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 . {\displaystyle \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}.}

These results can be substituted into the expression for ∇ p {\displaystyle \nabla p} to yield: ∇ p = − [ 1 μ 0 ∇ 2 A ] ∇ A − 1 μ 0 B z ∇ B z . {\displaystyle \nabla p=-\left\nabla A-{\frac {1}{\mu _{0}}}B_{z}\nabla B_{z}.}

Since p {\displaystyle p} and B z {\displaystyle B_{z}} are constants along a field line, and functions only of A {\displaystyle A} , hence ∇ p = d p d A ∇ A {\displaystyle \nabla p={\frac {dp}{dA}}\nabla A} and ∇ B z = d B z d A ∇ A {\displaystyle \nabla B_{z}={\frac {dB_{z}}{dA}}\nabla A} . Thus, factoring out ∇ A {\displaystyle \nabla A} and rearranging terms yields the Grad–Shafranov equation: ∇ 2 A = − μ 0 d d A ( p + B z 2 2 μ 0 ) . {\displaystyle \nabla ^{2}A=-\mu _{0}{\frac {d}{dA}}\left(p+{\frac {B_{z}^{2}}{2\mu _{0}}}\right).}

Derivation in contravariant representation

This derivation is only used for Tokamaks, but it can be enlightening. Using the definition of 'The Theory of Toroidally Confined Plasmas 1:3'(Roscoe White), Writing B → {\displaystyle {\vec {B}}} by contravariant basis ( ∇ Ψ , ∇ ϕ , ∇ ζ ) {\displaystyle (\nabla \Psi ,\nabla \phi ,\nabla \zeta )} : B → = ∇ Ψ × ∇ ϕ + F ¯ ∇ ϕ , {\displaystyle {\vec {B}}=\nabla \Psi \times \nabla \phi +{\bar {F}}\nabla \phi ,}

we have j → {\displaystyle {\vec {j}}} : μ 0 j → = ∇ × B → = − Δ ∗ Ψ ∇ ϕ + ∇ F ¯ × ∇ ϕ , where   Δ ∗ = r ∂ r ( r − 1 ∂ r ) + ∂ ϕ 2 ; {\displaystyle \mu _{0}{\vec {j}}=\nabla \times {\vec {B}}=-\Delta ^{*}\Psi \nabla \phi +\nabla {\bar {F}}\times \nabla \phi \quad {\text{, where}}\ \Delta ^{*}=r\partial _{r}(r^{-1}\partial _{r})+\partial _{\phi }^{2}{\text{;}}}

then force balance equation: μ 0 j → × B → = μ 0 ∇ p . {\displaystyle \mu _{0}{\vec {j}}\times {\vec {B}}=\mu _{0}\nabla p{\text{.}}}

Working out, we have: − Δ ∗ Ψ = F ¯ d F ¯ d Ψ + μ 0 R 2 d p d Ψ . {\displaystyle -\Delta ^{*}\Psi ={\bar {F}}{\frac {d{\bar {F}}}{d\Psi }}+\mu _{0}R^{2}{\frac {dp}{d\Psi }}{\text{.}}}

References

  1. Smith, S. G. L., & Hattori, Y. (2012). Axisymmetric magnetic vortices with swirl. Communications in Nonlinear Science and Numerical Simulation, 17(5), 2101-2107.

Further reading

  • Grad, H., and Rubin, H. (1958) Hydromagnetic Equilibria and Force-Free Fields Archived 2023-06-21 at the Wayback Machine. 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.
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