Grad - Shafranov equation

The Grad - Shafranov equation is the equation of plasma equilibrium in a tokamak . This equation was obtained by V. D. Shafranov in 1957 and independently by G. Grad and G. Rubin in 1958 .

In cylindrical coordinates, it has the form:

r^{2}\mathrm {div} \left({\frac {1}{r^{2}}}\nabla \Psi \right)=-r^{2}\mu _{0}\cdot p^{\prime }(\Psi )-{\frac {\mu _{0}^{2}}{4\pi ^{2}}}F(\Psi )\cdot F^{\prime }(\Psi )

{\ displaystyle r ^ {2} \ mathrm {div} \ left ({\ frac {1} {r ^ {2}}} \ nabla \ Psi \ right) = - r ^ {2} \ mu _ {0} \ cdot p ^ {\ prime} (\ Psi) - {\ frac {\ mu _ {0} ^ {2}} {4 \ pi ^ {2}}} F (\ Psi) \ cdot F ^ {\ prime } (\ Psi)}

{\ displaystyle r ^ {2} \ mathrm {div} \ left ({\ frac {1} {r ^ {2}}} \ nabla \ Psi \ right) = - r ^ {2} \ mu _ {0} \ cdot p ^ {\ prime} (\ Psi) - {\ frac {\ mu _ {0} ^ {2}} {4 \ pi ^ {2}}} F (\ Psi) \ cdot F ^ {\ prime } (\ Psi)}

,

or

{\frac {\partial ^{2}\Psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \Psi }{\partial r}}+{\frac {\partial ^{2}\Psi }{\partial z^{2}}}=-r^{2}\mu _{0}\cdot {\frac {dp}{d\Psi }}-{\frac {\mu _{0}^{2}}{4\pi ^{2}}}F(\Psi )\cdot {\frac {dF}{d\Psi }}

{\ displaystyle {\ frac {\ partial ^ {2} \ Psi} {\ partial r ^ {2}}} - {\ frac {1} {r}} {\ frac {\ partial \ Psi} {\ partial r }} + {\ frac {\ partial ^ {2} \ Psi} {\ partial z ^ {2}}} = - r ^ {2} \ mu _ {0} \ cdot {\ frac {dp} {d \ Psi}} - {\ frac {\ mu _ {0} ^ {2}} {4 \ pi ^ {2}}} F (\ Psi) \ cdot {\ frac {dF} {d \ Psi}}}

{\ displaystyle {\ frac {\ partial ^ {2} \ Psi} {\ partial r ^ {2}}} - {\ frac {1} {r}} {\ frac {\ partial \ Psi} {\ partial r }} + {\ frac {\ partial ^ {2} \ Psi} {\ partial z ^ {2}}} = - r ^ {2} \ mu _ {0} \ cdot {\ frac {dp} {d \ Psi}} - {\ frac {\ mu _ {0} ^ {2}} {4 \ pi ^ {2}}} F (\ Psi) \ cdot {\ frac {dF} {d \ Psi}}}

,

Where:

$\Psi$ ${\ displaystyle \ Psi}$ $\ Psi$ - magnetic flux through an external poloidal partition;
$F$ ${\ displaystyle F}$ $F$ - poloidal current;
$p$ ${\ displaystyle p}$ $p$ - plasma pressure;
$\mu _{0}$ ${\ displaystyle \ mu _ {0}}$ $\ mu _ {0}$ - magnetic constant .

Magnetic field induction:

{\vec {B}}={\frac {\mu _{0}\cdot F}{2\pi r}}{\vec {e}}_{\varphi }+{\frac {\nabla \Psi \times {\vec {e}}_{\varphi }}{r}}

{\ displaystyle {\ vec {B}} = {\ frac {\ mu _ {0} \ cdot F} {2 \ pi r}} {\ vec {e}} _ {\ varphi} + {\ frac {\ nabla \ Psi \ times {\ vec {e}} _ {\ varphi}} {r}}}

{\ displaystyle {\ vec {B}} = {\ frac {\ mu _ {0} \ cdot F} {2 \ pi r}} {\ vec {e}} _ {\ varphi} + {\ frac {\ nabla \ Psi \ times {\ vec {e}} _ {\ varphi}} {r}}}

Current density:

{\vec {j}}={\frac {1}{2\pi }}{\frac {dF}{d\Psi }}{\vec {B}}+{\frac {dp}{d\Psi }}r\cdot {\vec {e}}_{\varphi }

{\ displaystyle {\ vec {j}} = {\ frac {1} {2 \ pi}} {\ frac {dF} {d \ Psi}} {\ vec {B}} + {\ frac {dp} { d \ Psi}} r \ cdot {\ vec {e}} _ {\ varphi}}

{\ displaystyle {\ vec {j}} = {\ frac {1} {2 \ pi}} {\ frac {dF} {d \ Psi}} {\ vec {B}} + {\ frac {dp} { d \ Psi}} r \ cdot {\ vec {e}} _ {\ varphi}}

Literature

K.V. Brushlinsky , V.V. Saveliev . Magnetic traps to hold plasma. Mat. Modeling, vol. 11 N 5, 1999, pp. 3-36.
Physical Encyclopedia, T. 5, Toroidal Systems
TJM Boyd , JJ Sanderson The physics of plasmas
Kenrō Miyamoto Plasma physics and controlled nuclear fusion
Masahiro Wakatani Stellarator and heliotron devices

Grad - Shafranov equation

Literature

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