JOREK Reduced MHD Models

Refer also to the models, coordinates, notation, and normalization pages.

Note: The present JOREK model assumes that $F\equiv F_0$ is spatially and temporaly constant. The derivation shown here is valid for general $F(R,Z,t)$ which is, however, independent of $\phi$ to guarantee $\nabla\cdot\mathbf{B}=0$.

Definitions

Magnetic Field

The magnetic field is not normalized. A general form of $\mathbf{B}$ that satisfies $\nabla\cdot\mathbf{B}=0$ is given by:

\[\begin{align*} \mathbf{B} &= \nabla\psi\times \nabla\phi + F~\nabla\phi = \frac{1}{R}(-\partial_1\psi~\mathbf{a}^2 + \partial_2\psi~\mathbf{a}^1)+F\mathbf{a}^3 \\ \end{align*}\]

where $F$ and $\psi$ are scalars and $\partial_3 F=0$ to guarantee $\nabla\cdot\mathbf{B}=0$. The physical components (with respect to normalized basis vectors) are:

\[\begin{align*} B_R &= +\frac{1}{R}\frac{\partial\psi}{\partial Z} \\ B_Z &= -\frac{1}{R}\frac{\partial\psi}{\partial R} \\ B_\phi &= +\frac{F}{R} \end{align*}\]

The poloidal magnetic flux is connected with the vector potential by:

\[\begin{equation*} \psi = R^2\mathbf{A}\cdot\nabla\phi = R A_\phi \end{equation*}\]

Some additional remarks about the magnetic vector potential can be found here (not relevant for the derivation of the reduced MHD equations).

Current

The normalization of the plasma current is given by $\mathbf{j}_\text{SI}=\mathbf{j}/\mu_0$. The toroidal current density $j$ (it is not really a current density due to the factor R!) is defined as

\[\begin{equation*} j \equiv - R~\mathbf{j}\cdot\mathbf{e}_\phi = - \mathbf{j}\cdot\mathbf{a}_3 = \Delta^*\psi \end{equation*}\]

The derivation is sketched here. The physical component $j_\phi$ is therefore given by

\[\begin{equation*} j_\phi = -\frac{j}{R} = -\frac{\Delta^*\psi}{R} \end{equation*}\]

Although not represented by JOREK variables, R- and Z-components of the current exist as well (see here):

\[\begin{align*} j_R &= +\frac{1}{R}\partial_2 F+\frac{1}{R^2}\partial_{1,3}\psi \\ j_Z &= -\frac{1}{R}\partial_1 F+\frac{1}{R^2}\partial_{2,3}\psi \end{align*}\]

Velocity

Normalization (see also normalization table): $\mathbf{v}_\text{SI}=\mathbf{v}/\sqrt{\mu_0~\rho_0}$.

Standard Velocity

\[\begin{equation*} \begin{split} \mathbf{v} &= -R^2\nabla u \times \mathbf{a}^3 + v_{||}\mathbf{B} \\ &= R(\partial_1 u~\mathbf{a}^2 - \partial_2 u~\mathbf{a}^1) + v_{||}\left[\frac{1}{R}(-\partial_1\psi~\mathbf{a}^2 + \partial_2\psi~\mathbf{a}^1)+F\mathbf{a}^3\right] \\ &= \left(-R\partial_2 u+\frac{v_{||}}{R}\partial_2\psi\right)\mathbf{a}^1 + \left(R\partial_1 u-\frac{v_{||}}{R}\partial_1\psi\right)\mathbf{a}^2+F~v_{||}\mathbf{a}^3 \end{split} \end{equation*}\]

The physical components (with respect to normalized basis vectors) are:

\[\begin{align*} v_R &= -R\partial_2 u+\frac{v_{||}}{R}\partial_2\psi \\ v_Z &= +R\partial_1 u-\frac{v_{||}}{R}\partial_1\psi \\ v_\phi &= \frac{F~v_{||}}{R} \end{align*}\]

Ion Velocity with Diamagnetic Component

\[\begin{equation*} \begin{split} \mathbf{v} &= -R^2\nabla u \times \mathbf{a}^3 - \frac{\tau_{IC} R^2}{\rho}\nabla p_i\times\mathbf{a}^3 + v_{||}\mathbf{B} \\ &= R(\partial_1 u~\mathbf{a}^2 - \partial_2 u~\mathbf{a}^1) + v_{||}\left[\frac{1}{R}(-\partial_1\psi~\mathbf{a}^2 + \partial_2\psi~\mathbf{a}^1)+F\mathbf{a}^3\right] -\frac{\tau_{IC}R}{\rho} \left[ -\partial_1 p_i~\mathbf{a}^2 + \partial_2 p_i~\mathbf{a}^1 \right] \\ &= \left(-R\partial_2 u+\frac{v_{||}}{R}\partial_2\psi - \frac{\tau_{IC}R}{\rho}\partial_2 p_i\right)\mathbf{a}^1 + \left(R\partial_1 u-\frac{v_{||}}{R}\partial_1\psi + \frac{\tau_{IC}R}{\rho}\partial_1 p_i\right)\mathbf{a}^2+F~v_{||}\mathbf{a}^3 \end{split} \end{equation*}\]

(to be added soon…)

Vorticity

The toroidal component of the total vorticity $\mathbf{W}$ is:

\[\begin{equation*}\begin{split} \mathbf{W} \cdot \nabla \phi &= \nabla \cdot \left(\nabla_\text{pol}u +\frac{\tau_{IC}}{\rho} \nabla_\text{pol} p - v_{||} \frac{1}{R^2} \nabla_\text{pol} \psi \right) \end{split}\end{equation*}\]

(see here for the derivation).

The vorticity is normalized by $\mathbf{W}_\text{SI}=\mathbf{W}/\sqrt{\mu_0~\rho_0}$.

In the code, variable number 4, which we denote $\omega$, corresponds to the following part of the total vorticity:

\[\begin{equation*} \omega \equiv \nabla \cdot \nabla_\text{pol}u = \Delta_\text{pol} u \end{equation*}\]

Electric Potential

The velocity stream function $u$ is connected to the electric potential $\Phi$ by $\begin{equation*} u\equiv \Phi/F_0 \end{equation*}$ when $F\equiv F_0$ and the resistive term is small. The derivation is sketched here.

Useful Expressions

Expression	Description
$\nabla\times\nabla\phi\equiv 0$
$\nabla\cdot\nabla\phi\equiv 0$
$\nabla a\cdot\nabla\phi=\frac{1}{R^2}\partial_3 a$	For arbitrary scalars $a$, $b$
$\Delta_\text{pol} u=\Delta^* u+\frac{2}{R}\partial_1 u$
$\nabla a\times\nabla\phi = \frac{1}{R}(\partial_2 a\mathbf{a}_1 - \partial_1 a\mathbf{a}_2)$
$\nabla\phi\times(\nabla a\times\nabla\phi)=\frac{1}{R^2}\nabla_\text{pol}a$
$(\nabla a\times\nabla\phi)\times(\nabla b\times\nabla\phi) = \frac{1}{R}[a,b]\nabla\phi$
$\nabla\times(\nabla a\times\nabla\phi)=-\Delta^* a\nabla\phi + \frac{1}{R^2}\nabla_\text{pol}\partial_3 a$
$\mathbf{B}\cdot\nabla a=\frac{F}{R^2}\partial_3 a+\frac{1}{R}[a,\psi]$
$\nabla a\cdot\nabla b = \nabla_\text{pol}a\cdot\nabla_\text{pol}b+\frac{1}{R^2}\partial_3 a\partial_3 b$
$\nabla\phi\cdot(\nabla C\times \nabla D)=\frac{1}{R}[C,D]$	For arbitrary vectors $\mathbf{C}$, $\mathbf{D}$
$\mathbf{B}\cdot(\mathbf{C}\times\mathbf{D})=\frac{F}{R}[C,D]-\frac{C}{R}[\nabla\psi,D]+\frac{D}{R}[\nabla\psi,C]$
$\mathbf{j} \equiv \nabla F\times\nabla\phi + \frac{1}{R^2}\partial_3(\nabla_\text{pol}\psi)-\Delta^*\psi~\nabla\phi$	see here
$j \equiv -\mathbf{j}\cdot\mathbf{a}_3=\Delta^*\psi$	see here
$\mathbf{j}\times\mathbf{B}= -\frac{F}{R^2}\nabla_\text{pol}F+\frac{1}{R}\nabla\phi[F,\psi]-\frac{\Delta^*\psi}{R^2}\nabla_\text{pol}\psi+\frac{1}{R^2}\nabla_\text{pol}\partial_3\psi\times\mathbf{B}$	see here
$\mathbf{v}\times\mathbf{B} = \frac{1}{R}\mathbf{a}3 [\psi,u]+F\nabla\text{pol} u$	see here

Equations

Continuity Equation

\[\begin{equation} \partial_t\rho=-\nabla\cdot(\rho\mathbf{v})+\nabla\cdot(D\nabla\rho) + S_\rho \end{equation}\]

The derivation in reduced MHD form is given here.

Poloidal Flux Equation

\[\begin{equation} \partial_t\psi=R[\psi,u]+\eta(T)(j-j_0)-F_0\partial_\phi u \end{equation}\]

where $j_0$ is implemented acting as a current source. The derivation is sketched here.

Perpendicular Momentum Equation

The perpendicular projection of the momentum equation is obtained by applying the operator $\nabla\phi\cdot\nabla\times(R^2…)$ onto the momentum equation.

Standard Equation

\[\begin{equation*}\begin{split} \nabla\cdot\left[R^2\rho\nabla_{pol}\left(\frac{\partial U}{\partial t}\right)\right] = & \frac{1}{R} \left[R^4\rho\omega,U\right] - \frac{1}{2R} \left[R^2\rho,R^2\left|\nabla_{pol}U\right|^2\right] \\ & - \frac{1}{R} \left[R^2,p\right] + \frac{1}{R} \left[\psi,j\right] - \frac{F_{o}}{R^2}\frac{\partial j}{\partial\phi} + \mu \nabla^2\omega \end{split}\end{equation*}\]

Equation with Diamagnetic Components

This equation is obtained by replacing $\mathbf{v}_{pol} = R^2 \nabla\phi\times\nabla U$ with $\mathbf{v}_{pol} = R^2\nabla\phi\times\nabla U + \frac{\tau_{IC}R^2}{\rho}\nabla\phi\times\nabla P_i$ and taking out the terms that cancel out with the gyro-viscous tensor (see gyro-viscous cancellation, [Schnack, Phys. Plasmas 13, 058103, 2006]).

\[\begin{equation*}\begin{split} \nabla\cdot\left[R^2\rho\nabla_{pol}\left(\frac{\partial U}{\partial t}\right)\right] = & \frac{1}{R} \left[R^4\rho W,U\right] - \frac{1}{2R} \left[R^2\rho,R^2\left|\nabla_{pol}U\right|^2\right] \\ & - \frac{1}{R} \left[R^4\tau_{IC}\nabla^{2}_{pol}P_{i},U\right] + \frac{1}{R} \left[R^4\frac{\tau_{IC}}{\rho}\nabla_{pol}\rho\cdot\nabla_{pol}P_{i},U\right] \\ & + \tau_{IC}R^3\left[W,P_i\right] + \tau_{IC}R^2 \nabla\cdot\left(\frac{\partial P_i}{\partial Z}\nabla_{pol}U\right) \\ & - \tau_{IC}R^3 \left[\partial_{xy}U \left(\partial_{xx}P_i-\partial_{yy}P_i\right) -\partial_{xy}P_i\left(\partial_{xx}U -\partial_{yy}U \right)\right] \\ & - \frac{1}{R} \left[R^{2},p\right] + \frac{1}{R} \left[\psi,j\right] - \frac{F_{o}}{R^2}\frac{\partial j}{\partial\phi} + \mu \nabla^{2}\left(\omega + \omega^*\right) \end{split}\end{equation*}\]

where $\omega^$ is the diamagnetic part of the vorticity (see definition here). Note that it is very important to include $\omega^$ in the equation. In the code, we choose to define the total vorticity as

\[\begin{equation*} W = \omega + \omega^* \end{equation*}\]

Note that introducing this new variable in the above equation means we have to extract $\omega^$ from the term $\left[R^4\rho\omega,U\right]$. Ideally, we would also need to extract $\omega^$ from the term $\tau_{IC}R^4[\omega,P_i]$, but this would result in a term of order $\tau_{IC}^2$, which is considered small enough to be ignored. The resulting equation is (with the new definition of the total vorticity $W$):

Parallel Momentum Equation

The parallel projection of the momentum equation is obtained by applying the operator $\nabla\phi\cdot (…)$ onto the momentum equation.

The derivation can be found here.

Standard Equation

\[\begin{equation*}\begin{split} \rho \frac{\partial v_{\parallel}}{\partial t} = & - R \rho \left[ u, v_{\parallel}\right] + \frac{1}{R} \rho v_{\parallel} \left[ \psi, v_{\parallel}\right] - \frac{F_0}{R^2} \rho v_{\parallel} \partial_3 v_{\parallel} \\ & - \frac{1}{2R^2} \frac{1}{F_0} \frac{\partial}{\partial \phi} \left| \nabla \psi\right|^2 - \frac{1}{F_0} \frac{\partial p}{\partial \phi} \end{split}\end{equation*}\]

Energy Equation

(To be filled soon…)

Current Definition Equation

(To be filled soon…)

Vorticity Definition Equation

(To be filled soon…)